Measure Integral and Real Analysis (1)

\(X / A\) means that the difference between set \(X\) and \(A\).

Fields

Complete Ordered Fields

Definition 0.1: Field

A field is a set \(\mathbb{F}\) along with closed operations of addition and multiplication on \(\mathbb{F}\) that have the following properties:

1. Commutativity: \[a + b = b + a, \quad ab = ba \quad \forall a, b \in \mathbb{F}\]
2. Associativity: \[(a + b) + c = a + (b + c) \quad (ab)c = a(bc) \quad \forall a,b,c \in \mathbb{F}\]
3. Distributive Property: \[a(b + c) = ab + ac \quad \forall a, b, c \in \mathbb{F}\]
4. Additive Identity: There exists an element \(0 \in \mathbb{F}\) s.t. \(a + 0 = a, \forall a \in \mathbb{F}\)
5. Additive Inverse: For each \(a \in \mathbb{F}\), there exists an element \(-a \in \mathbb{F}\) such that \(a + (-a) = 0\).
6. Multiplicative Identity: There exists an element \(1 \in \mathbb{F}\) s.t \(1 \neq 0\) and \(a1 = a, \forall a \in \mathbb{F}\).
7. Multiplicative Inverse: For each \(a \in \mathbb{F}\) with \(a \neq 0\), there exists an element \(a^{-1} \in \mathbb{F}\) s.t \(aa^{-1} = 1\)

\(\mathbb{Q}, \mathbb{R}, \mathbb{C}, \{0, 1\}\) with usual operation of addition and multiplication are fields.

Definition 0.5: Ordered Field, Positive

An ordered field is a field \(\mathbb{F}\) along with a subset \(P\) of \(\mathbb{F}\), called the positive subset with the following properties:

1. If \(a \in \mathbb{F}\), then \(a \in P\) or \(a = 0\) or \(-a \in P\).
2. If \(a \in P\), then \(-a \notin P\).
3. If \(a, b \in P\), then \(a + b \in P\) and \(ab \in P\).

Theorem 0.6: The Positive Subset is Closed under Multiplicative Inverse

Suppose \(\mathbb{F}\) is an ordered field with positive subset \(P\). Then:

1. \(1 \in P\)
2. \(a^{-1} \in P, \forall a \in P\)

Definition 0.7: Less Than, Greater Than

Suppose \(\mathbb{F}\) is an ordered field with positive subset \(P\). Suppose \(a, b \in \mathbb{F}\). Then:

1. \(a < b\) is defined to mean \(b - a \in P\).
2. \(a \leq b\) is defined to mean \(a < b\) or \(a = b\).
3. \(a > b\) is defined to mean \(a - b \in P\).
4. \(a \geq b\) is defined to mean \(a > b\) or \(a = b\).

The statement \(0 < b\) is equivalent to the statement \(b \in P\).

Theorem 0.8: Transitivity

Suppose \(\mathbb{F}\) is an ordered field and \(a, b, c \in \mathbb{F}\). If \(a < b\) and \(b < c\), then \(a < c\).

Definition 0.9: Absolute Value

Suppose \(\mathbb{F}\) is an ordered field and \(b \in \mathbb{F}\). The absolute value of \(b\), denoted \(|b|\), is defined by:

\[ |b| = \begin{cases} b, \quad \text{if} b \geq 0\\ -b, \quad \text{if} b < 0 \end{cases} \]

Theorem 0.10: \(|a + b| \leq |a| + |b|\)

Suppose \(\mathbb{F}\) is an ordered field and \(a, b \in \mathbb{F}\). Then:

\[|a + b| \leq |a| + |b|\]

Completeness

Definition 0.19: Complete Ordered Field

An ordered field \(\mathbb{F}\) is called complete if every nonempty subset of \(\mathbb{F}\) that has an upper bound has a least upper bound.

Definition 0.20: \(\mathbb{R}\), The Field of Real Numbers

The symbol \(\mathbb{R}\) denotes a complete ordered field. This field is called real numbers.

Definition 0.35: Supremum and Infimum

Suppose \(A \subseteq \mathbb{R}\). The supremum of \(A\), denoted \(\sup A\), is defined as:

\[ \sup A = \begin{cases} \text{The Least Upper Bound}, \quad \text{if $A$ has an upper bound and $A \neq \emptyset$}\\ \infty, \quad \text{if $A$ does not have an upper bound}\\ -\infty, \quad \text{if $A = \emptyset$} \end{cases} \]

The Infimum of \(A\), denoted \(\inf A\), is defined as:

\[ \inf A = \begin{cases} \text{The Greatest Lower Bound}, \quad \text{if $A$ has a lower bound and $A \neq \emptyset$}\\ -\infty, \quad \text{if $A$ does not have a lower bound}\\ \infty, \quad \text{if $A = \emptyset$} \end{cases} \]

Intervals

Sometimes it is useful to consider a set consisting of \(\mathbb{R}\) and two additional elements called \(\infty\) and \(-\infty\). We define it as \(\mathbb{R} \cup \{\infty, -\infty\}\).

Definition 0.40: Ordering on \(\mathbb{R} \cup \{\infty, -\infty\}\)

• The ordering \(<\) on \(\mathbb{R}\) is extended to \(\mathbb{R} \cup \{\infty, -\infty\}\) as follows:
  • \(a < \infty, \;\forall a \in \mathbb{R} \cup \{-\infty\}\)
  • \(-\infty < a, \;\forall a \in \mathbb{R} \cup \{\infty\}\)
• For \(a, b \in \mathbb{R}\cup \{\infty, -\infty\}\),
  • The notation \(a \leq b\) means that \(a < b\) or \(a = b\).
  • The notation \(a > b\) means that \(b < a\).
  • The notation \(a \geq b\) means that \(a > b\) or \(a = b\).

Definition 0.41: Interval Notation

Suppose \(a, b \in \mathbb{R} \cup \{\infty, -\infty\}\). Then:

• \((a, b) = \{t\in\mathbb{R}: a < t < b\}\)
• \([a, b] = \{t \in \mathbb{R} \cup \{\infty, -\infty\}: a \leq t \leq b\}\)
• \((a, b] = \{t \in \mathbb{R} \cup \{\infty\}: a < t \leq b\}\)
• \([a, b) = \{t \in \mathbb{R} \cup \{-\infty\}: a \leq t < b\}\)

If \(a > b\) then all sets are empty ses. If \(a = b\), then \([a, b]\) is the set of \(\{a\}\), all others are empty sets. The definition implies that:

1. \((-\infty, \infty) = \mathbb{R}\)
2. \([-\infty, \infty] = \mathbb{R} \cup \{\infty, \infty\}\), this is not a subset of \(\mathbb{R}\)

Definition 0.42: Interval

• A subset of \([-\infty, \infty]\) is called an interval if it contains all numbers that are between pairs of its elements.
• In other words, a set \(I \subset [-\infty, \infty]\) is called an interval if \(c, d \in I\) implies \((c, d) \in I\).

Theorem 0.43: Description of Intervals

Suppose \(I \subset [-\infty, \infty]\) is an interval. Then \(I\) is one of the following sets for some \(a,b \in [-\infty, \infty]\):

\[(a, b), [a, b], (a, b], [a, b)\]

Open and Closed Subsets of \(\mathbb{R}^n\)

Definition 0.44: \(\mathbb{R^n}\)

^n is the set of all ordered \(n-tuples\) of real numbers:

\[\mathbb{R}^n = \{(x_1, ..., x_n): x_1, ..., x_n \in \mathbb{R}\}\]

Definition 0.45: \(\|\cdot\|, \|\cdot\|_{\infty}\)

For \((x_1, ..., x_n) \in \mathbb{R}^n\), let:

\[\|(x_1, ..., x_n)\| = \sqrt{x_1^2 + ... + x_n^2}\]

and

\[\|(x_1, ..., x_n)\|_{\infty} = \max\{|x_1|, ..., |x_n|\}\]

Definition 0.46: Limit

Suppose \(a_1, a_2, ... \in\mathbb{R}^n\) and \(L \in \mathbb{R}^n\). Then \(L\) is called a limit of the sequence \(a_1, a_2, ...\) and we write:

\[\lim_{k \rightarrow \infty} a_k = L\]

If for every \(\epsilon > 0\), there exists \(m \in \mathbb{Z}^+\) s.t:

\[\|a_k - L\|_\infty < \epsilon\]

for all integers \(k \geq m\).

Definition 0.47: Converge, Convergent

A sequence in \(\mathbb{R}^n\) is said to converge and to be a convergent sequence if it has a limit.

Theorem 0.48: Coordinatewise Limits

Suppose \(a_1, a_2, .... \in \mathbb{R}^n\) and \(L \in \mathbb{R}^n\). For \(k \in \mathbb{Z}^+\), let:

\[(a_{k, 1}, ...., a_{k, n}) = a_k\]

and let \((L_1, ..., L_n) = L\). Then, \(\lim_{k \rightarrow \infty} a_k = L\) IFF:

\[\lim_{k \rightarrow \infty} a_{k, j} = L_j\]

For each \(j \in \{1, ..., n\}\)

Thus, questions about convergence of sequences in \(\mathbb{R}^n\) can often be reduced to questions about convergence of sequences in \(\mathbb{R}\).

Definition 0.49: Open Cube

For \(x \in \mathbb{R}^n\) and \(\delta > 0\), the open cube \(B(x, \delta)\) is defined by:

\[B(x, \delta) = \{y \in \mathbb{R}^n: \|y - x\|_{\infty} < \delta\}\]

Definition 0.51: Open Interval

A subset of \(\mathbb{R}\) of the form \((a, b)\) for some \(a, b \in [-\infty, \infty]\) is called an open interval.

If \(n = 1\), then \(B(x, \delta) = (x - \delta, x + \delta)\)

Definition 10.52: Open Subset of \(\mathbb{R}^n\)

• A subset \(G\) of \(\mathbb{R}^n\) is called open if for every \(x \in G\), there exists \(\delta > 0\) s.t \(B(x, \delta) \subseteq G\).
• Equivalently, a subset \(G\) of \(\mathbb{R}^n\) is called open if every element of \(G\) is contained in an open cube that is contained in \(G\).

Theorem 0.55: Union and Intersection of Open Sets

1. The union of every collection of open subsets of \(\mathbb{R}^n\) is an open subset of \(\mathbb{R}^n\).
2. The intersection of every finite collection of open subset of \(\mathbb{R}^n\) is an open subset of \(\mathbb{R}^n\).

Definition 0.56: Countable, Uncountable

• A set \(C\) is called countable if \(C = \emptyset\) or if \(C = \{c_1 c_2, ...\}\) for some sequence \(c_1, c_2, ...\) of element of \(C\).
• A set is called uncountable if it is not countable.

Definition 0.58: Disjoint

A sequence \(E_1, E_2, ...\) of sets is called disjoint if \(E_j \cap E_k = \emptyset\) whenever \(j \neq k\).

Theorem 0.59: Open Subset of \(\mathbb{R}\) is Countable Disjoint Union of Open Intervals

A subset of \(\mathbb{R}\) is open iFF it is the union of a disjoint sequence of open interval.

Definition 0.60: Set Difference, Complement

• If \(S, A\) are sets, then the set difference \(S \ A\) is defined to be the set of elements of \(S\) that are not in \(A\). In other words, \(S \ A = \{s \in S: s \neq A\}\).
• If \(A \subseteq S\), then \(S \ A\) is the complement of \(A\) in \(S\).

Definition 0.61: Closed Subset of \(\mathbb{R}^n\)

A subset of \(\mathbb{R}^n\) is called closed if its complement in \(\mathbb{R}^n\) is open.

a subset of \(\mathbb{R}^n\) need not be either open or closed. For example \((6, 16]\) is neither open nor closed

Theorem 0.62: Characterization of Closed Sets

A subset of \(\mathbb{R}^n\) is closed IFF it contains the limit of every convergent sequence of elements of the set.

Theorem 0.63: De Morgan's Laws

Suppose \(A\) is a collection of subsets of some set \(X\). Then:

\[X /\ \bigcup_{E \in A} E = \bigcap_{E \in A} (X /\ E)\]

and

\[X /\ \bigcap_{E \in A} E = \bigcup_{E \in A} (X /\ E)\]

Theorem 0.65: Sets that are Both Open and Closed

The only subsets of \(\mathbb{R}^n\) that are both open and closed are \(\emptyset\) and \(\mathbb{R}^n\).

Riemann Integral

Riemann Integral Review

Definition 1.1: Partition

Suppose \(a, b \in \mathbb{R}\) with \(a < b\). A partition of \([a, b]\) is a finite list of the form \(x_0, x_1, ...., x_n\), where:

\[a = x_0 < x_1 < .... < x_n = b\]

Definition 1.2: Notation for Infimum and Supremum of a Function

If \(f\) is a real-valued function and \(A\) is a subset of the domain of \(f\), then:

\[\inf_A f = \inf\{f(x): x \in A\} \quad \quad \sup_A f = \sup\{f(x): x \in A\}\]

Definition 1.3: Lower and Upper Riemann Sums

Suppose \(f: [a, b] \rightarrow \mathbb{R}\) is a bounded function and \(P\) is a partition \(x_0 ,...., x_n\) of \([a, b]\). The lower Riemann sum \(L(f, P, [a, b])\) and the upper Riemann sum \(U(f, P, [a, b])\) are defined by:

\[L(f, P, [a, b]) = \sum^n_{j=1}(x_j - x_{j-1}) \inf_{[x_{j-1}, x_j]} f\]

and

\[U(f, P, [a, b]) = \sum^n_{j=1}(x_j - x_{j-1}) \sup_{[x_{j-1}, x_j]} f\]

Theorem 1.5: Inequalities with Riemann Sums

Suppose \(f: [a, b] \rightarrow \mathbb{R}\) is a bounded function and \(P, P^{\prime}\) are partitions of \([a, b]\), such that the list defining \(P\) is a sublist of the list defining \(P^{\prime}\). Then:

\[L(f, P, [a, b]) \leq L(f, P^{\prime}, [a, b]) \leq U(f, P^{\prime}, [a, b]) \leq U(f, P, [a, b])\]

Proof of Theorem 1.5:

Let \(P\) be the partition \(x_0, ..., x_n\) and \(P^{\prime}\) be the partition \(x^{\prime}_0, ..., x^{\prime}_N\) of \([a, b]\). Then for each \(j = 1, ..., n\), there exists \(k \in \{0, ..., N-1\}\) and a positive integer \(m\) s.t \(x_{j-1} = x^{\prime}_k < ... < x^{\prime}_{k+m} = x_j\), in other words, the interval \([x_{j-1}, x_j]\) contains several smaller intervals \([x^{\prime}_{k}, x^{\prime}_{k+1}], ...., [x^{\prime}_{k+m -1}, x^{\prime}_{k+m}]\). Then:

\[(x_j - x_{j-1}) \inf_{[x_{j-1}, x_j]} f = \sum^{m}_{i=1} (x^{\prime}_{k+i} - x^{\prime}_{k+i - 1})\inf_{[x_{j-1}, x_j]} f \leq \sum^{m}_{i=1} (x^{\prime}_{k+i} - x^{\prime}_{k+i - 1})\inf_{[x^{\prime}_{k+i - 1}, x^{\prime}_{k+i}]} f\]

Which implies the first inequality. The second and third are similar.

Theorem 1.6: Lower Riemann Sums \(\leq\) Upper Riemann Sums

Suppose \(f: [a, b] \rightarrow \mathbb{R}\) is a bounded function and \(P, P^{\prime}\) are partitions of \([a, b]\). Then:

\[L(f, P, [a, b]) \leq U(f, P^{\prime}, [a, b])\]

Proof of Theorem 1.6:

Let \(P^{\prime\prime}\) be the partition of \([a, b]\) obtained by merging the lists that define \(P, P^{\prime}\), then by theorem 1.5, we have:

\[L(f, P, [a, b]) \leq L(f, P^{\prime\prime}, [a, b]) \leq U(f, P^{\prime\prime}, [a, b]) \leq U(f, P^{\prime}, [a, b])\]

For any \(P, P^{\prime}\).

Definition 1.7: Lower and Upper Riemann Integrals

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is a bounded function. The lower Riemann integral \(L(f, [a, b])\) and the upper Riemann integral \(U(f, [a, b])\) of \(f\) are defined by:

\[L(f, [a, b]) = \sup_{P} L(f, P, [a, b])\]

and

\[U(f, [a, b]) = \inf_P U(f, P, [a, b])\]

Where the supremum and infimum above are taken over all partitions \(P\) of \([a, b]\).

Theorem 1.8: Lower Riemann Integral \(\leq\) Upper Riemann Integral

Suppose \(f: [a, b] \rightarrow \mathbb{R}\) is a bounded function. Then:

\[L(f, [a, b]) \leq U(f, [a, b])\]

Definition 1.9: Riemann Integrable; Riemann Integral

• A bounded function on a closed bounded interval is called Riemann integrable if its lower Riemann integral equals its upper Riemann integral.
• If \(f: [a, b] \rightarrow \mathbb{R}\) is Riemann integrable, then the Riemann integral \(\int^a_b f\) is defined by: \[\int^b_a f = L(f, [a, b]) = U(f, [a, b])\]

Theorem 1.11: Continuous Functions are Riemann Integrable

Every continuous real-valued function on each closed bounded interval is Riemann integrable.

Theorem 1.13: Bounds on Riemann Integral

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is Riemann integrable. Then:

\[(b - a) \inf_{[a, b]}f \leq \int^b_a f \leq (b - a) \sup_{[a, b]} f\]

Riemann Integral Is not Good Enough

The Riemann integral has several deficiencies:

1. Riemann integration does not handle functions with many discontinuities.
2. Riemann integration does not handle unbounded functions.
3. Riemann integration does not work well with limits.

Measures

Outer Measure on \(\mathbb{R}\)

Definition 2.1: Length of Open Interval: \(l(I)\)

The length \(l(I)\) of an open interval \(I\) is defined by:

\[ l(I) = \begin{cases} b - a, \quad \text{if $I = (a, b)$ for some $a, b \in \mathbb{R}$ with $a < b$}\\ 0, \quad \text{if $I = \emptyset$}\\ \infty, \quad \text{if $I = (-\infty, a)$ or $I = (a, \infty)$ for some $a \in \mathbb{R}$}\\ \infty, \quad \text{if $I = (-\infty, \infty)$} \end{cases} \]

Definition 2.2: Outer Measure: \(|A|\)

The outer measure \(|A|\) of a set \(A \subseteq \mathbb{R}\) is defined by:

\[|A| = \inf\{\sum^{\infty}_{k=1} l(I_k): I_1, I_2, ... \text{ are open intervals such that $A \subseteq \bigcup_{k=1}^{\infty} I_k$}\}\]

Example 2.3: Finite sets have outer measure 0

Suppose \(A = \{a_1, ..., a_n\}\) is a finite set of real numbers. Suppose \(\epsilon > 0\). Define a sequence \(I_1, I_2, ...\) of open intervals by:

\[ I_k = \begin{cases} (a_k - \epsilon, a_k + \epsilon), \quad \text{if $k \leq n$}\\ \emptyset, \quad \text{if $k > n$}\\ \end{cases} \]

Then \(I_1, I_2, ...\) is a sequence of open interval whose union contains \(A\). Clearly \(\sum^{\infty}_{k=1}l(I_k) = 2\epsilon n \implies |A| \leq 2\epsilon n\), since \(\epsilon > 0\) is an arbitrary positive number, this implies that \(|A| = 0\).

Theorem 2.4: Countable Sets Have Outer Measure \(0\)

Every countable subset of \(\mathbb{R}\) has outer measure \(0\).

Theorem 2.5: Outer Measure Preserves Order

Suppose \(A, B\) are subsets of \(\mathbb{R}\) with \(A \subseteq B\). Then \(|A| \leq |B|\).

Definition 2.6: Translation, \(t + A\)

If \(t \in \mathbb{R}\) and \(A \subseteq \mathbb{R}\), then the translation \(t + A\) is defined by:

\[t + A = \{t + a: a \in A\}\]

If \(t \in \mathbb{R}, a, b \in [-\infty, \infty]\), then \(t + (a, b) = (t + a, t + b)\) and \(t + \infty = \infty\) and \(t + -\infty = -\infty\).

Theorem 2.7: Outer Measure is Translation Invariant

Suppose \(t \in \mathbb{R}\) and \(A \subseteq \mathbb{R}\). Then \(|t + A| = |A|\).

Theorem 2.8: Countable Subadditivity of Outer Measure

Suppose \(A_1, A_2, ...\) is a sequence of subsets of \(\mathbb{R}\). Then:

\[|\bigcup_{k=1}^{\infty} A_k| \leq \sum^{\infty}_{k=1} |A_k|\]

In other words:

\[|A_1 \cup A_2 \cup ... | \leq |A_1| + |A_2| + ... \]

Proof of Theorem 2.8

Assume \(|A_k| < \infty\) for all \(k \in \mathbb{Z}^+\). Let \(\epsilon > 0\) and for each \(k \in \mathbb{Z}^+\), let \(I_{1, k}, I_{2, k}, ...\) be a sequence of open intervals whose union contains \(A_k\) s.t:

\[\sum^{\infty}_{j=1} l(I_{j, k}) \leq \frac{\epsilon}{2^k} + |A_k|\]

This makes sense because \(|A| = \inf\{\sum^{\infty}_{j=1} l(I_{j, k})\}\) and \(|A| + \epsilon \geq \inf\{\sum^{\infty}_{j=1} l(I_{j, k})\}\).

Sum both side over \(k\), we have:

\[\sum^{\infty}_{k=1}\sum^{\infty}_{j=1} l(I_{j, k}) \leq \epsilon + \sum^{\infty}_{k=1}|A_k|\]

The union of left hand side open intervals is the set $A_1 A_2 ... $, Thus, we have:

\[|A_1 \cup A_2 \cup ... | \leq \sum^{\infty}_{k=1}\sum^{\infty}_{j=1} l(I_{j, k}) \leq \epsilon + \sum^{\infty}_{k=1}|A_k|\]

Since \(\epsilon\) is any positive real number, we have desired result.

Definition 2.10: Open Cover, Finite Subcover

Suppose \(A \subseteq \mathbb{R}\):

• A collection \(C\) of open subsets of \(\mathbb{R}\) is called an open cover of \(A\) if \(A\) is contained in the union of all the sets in \(C\).
• An open cover \(C\) of \(A\) is said to have a finite subcover if \(A\) is contained in the union of some finite list of sets in \(C\).

Example 2.11

The collection \(\{(k, k+2): k \in \mathbb{Z}^+\}\) is an open cover of \([2, 5]\) because \([2, 5] \subseteq \bigcup^{\infty}_{k=1} (k, k+2)\). This open cover has finite subcover because \([2, 5] \subseteq (1, 3) \cup (2, 4) \cup (3, 5) \cup (4, 6)\).

The collection above is an open cover of \([2, \infty)\) but does not have a finite subcover.

Theorem 2.12: Heine-Borel Theorem

Every open cover of a closed bounded subset of \(\mathbb{R}\) has a finite subcover.

Theorem 2.14: Outer Measure of a Closed Interval

Suppose \(a, b \in \mathbb{R}\) with \(a < b\). Then \(|[a, b]| = b - a\)

Theorem 2.17: Nontrivial Intervals are Uncountable

Every interval in \(\mathbb{R}\) that contains at least two distinct elements is uncountable.

Proof of Theorem 2.17:

Suppose \(I\) is an interval that contains \(a, b \in \mathbb{R}\) and \(b > a\), then \([a, b] \subseteq I\), then by theorem 2.5, we have:

\[|I| \geq |[a, b]|\]

and

\[|[a, b]| = b - a > 0\]

Theorem 2.18: Nonadditivity of Outer Measure

There exist disjoint subsets \(A, B\) of \(\mathbb{R}\) s.t:

\[|A \cup B| \neq |A| + |B|\]

Measurable Spaces and Functions

Theorem 2.22: Nonexistence of Extension of Length to All Subsets of \(\mathbb{R}\)

There does not exists a function \(\mu\) with all the following properties:

1. \(\mu\) is a function from the set of subsets of \(\mathbb{R}\) to \([0, \infty]\).
2. \(\mu(I) = l(I)\) for every open interval \(I\) of \(\mathbb{R}\).
3. \(\mu (\sum^{\infty}_{k=1} A_k) = \sum^{\infty}_{k=1} \mu(A_k)\) for every disjoint sequence \(A_1, A_2, ...\) of subsets of \(\mathbb{R}\).
4. \(\mu(t + A) = \mu(A)\) for every \(A \subseteq \mathbb{R}\) and every \(t \in \mathbb{R}\).

Definition 2.23: \(\sigma\)-algebra

Suppose \(X\) is a set and \(S\) is a set of subsets of \(X\). Then \(S\) is called a \(\sigma\)-algebra on \(X\) if the following three conditions are satisfied:

• \(\emptyset \in S\).
• If \(E \in S\), then \(X / E \in S\).
• If \(E_1, E_2, ...\) is a sequence of elements of \(S\), then \(\bigcup^{\infty}_{k=1} E_k \in S\).

{, X} is \(\sigma\)-algebra on \(X\). The set of all subsets of \(X\) is \(\sigma\)-algebra on \(X\).

Theorem 2.25: \(\sigma\)-algebras are Closed Under Countable Intersection

Suppose \(S\) is a \(\sigma\)-algebra on a set \(X\). Then:

1. \(X \in S\).
2. If \(D, E \in S\), then \(D \cup E \in S\) and \(D \cap E \in S\) and \(D / E \in S\).
3. If \(E_1, E_2, ....\) is a sequence of elemnts of \(S\), then \(\bigcap^{\infty}_{k=1} E_k \in S\).

Definition 2.26: Measurable Space, Measurable Set

• A measurable space is an ordered pair \((X, S)\), where \(X\) is a set and \(S\) is a \(\sigma-\)algebra on \(X\).
• An element of \(S\) is called an \(S\)-measurable set or just a measurable set if \(S\) is clear from the context.

\(X = \mathbb{R}\) and \(S\) is the set of all subsets of \(\mathbb{R}\) that are countable or have a countable complement, then the ordered pair \((X, S)\) is a measurable space and \(\mathbb{Q} \in S\) is \(S\)-measurable set.

Theorem 2.27: Smallest \(\sigma\)-algebra Containing a Collection of Subsets

Suppose \(X\) is a set and \(A\) is a set of subsets of \(X\). Then the intersection of all \(\sigma-\)algebras on \(X\) that contain \(A\) is a \(\sigma-\)algebra on \(X\).

Using the smallest for the intersection of all \(\sigma\)-algebras that contain as set \(A\) of subset of \(X\) makes sense, because the smallest \(\sigma\)-algebra will be one of those \(\sigma\)-algebra that contains \(A\). If \(A\) is already \(\sigma\)-algebra, then the interaction is just \(A\).

Definition 2.29: Borel Set

The smallest \(\sigma\)-algebra on \(\mathbb{R}\) containing all open subsets of \(\mathbb{R}\) is called the collection of Borel subsets of \(\mathbb{R}\). An element of this \(\sigma-algebra\) is called a Borel set.

We can also define the collection of Borel subsets to be the smallest \(\sigma\)-algebra on \(\mathbb{R}\) containing all the open intervals. Because every open subset of \(\mathbb{R}\) is the union of a sequence of open intervals.

The set contains every open sets, closed sets, countable union, intersection of open, closed sets. There exists subsets of \(\mathbb{R}\) that are not Borel sets, but any subset of \(\mathbb{R}\) that you can write down in a concrete fashion is a Borel set.

Definition 2.31: Inverse Image: \(f^{-1} (A)\)

If \(f: X \rightarrow Y\) is a function and \(A \subseteq Y\), then the set \(f^{-1} (A)\) is defined by:

\[f^{-1} (A) = \{x \in X: f(x) \in A\}\]

Theorem 2.33: Algebra of Inverse Images

Suppose \(f: X \rightarrow Y\) is a function. Then:

1. \(f^{-1} (Y / A) = X / f^{-1}(A), \; \forall A \subseteq Y\)
2. \(f^{-1} (\bigcap_{A \in \mathbf{A}} A) = \bigcap_{A \in \mathbb{A}} f^{-1}(A)\), for every set \(\mathbf{A}\) of subsets of \(Y\).
3. \(f^{-1} (\bigcup_{A \in \mathbf{A}} A) = \bigcup_{A \in \mathbb{A}} f^{-1}(A)\), for every set \(\mathbf{A}\) of subsets of \(Y\).

In other words, the inverse image of union or intersection of any subsets of \(Y\), is the same as the union or intersection of inverse image of these subsets of \(Y\).

Proof of Theorem 2.33:

Suppose \(A \subseteq Y\). For \(x \in X\), we have:

\[x \in f^{-1} (Y / A) \Longleftrightarrow f(x) \in Y / A \Longleftrightarrow f(x) \notin A \Longleftrightarrow x \notin f^{-1}(A) \implies x \in X / f^{-1}(A) \]

Let \(\mathbf{A}\) be a set of subsets of \(Y\), then:

\[x \in f^{-1} (\bigcap_{A \in \mathbf{A}} A) \Longleftrightarrow f(x) \in \bigcap_{A \in \mathbf{A}} A\]

This implies that \(f(x)\) is in every subset \(A \in \mathbf{A}\), thus, we have:

\[x \in f^{-1}(A) \; \forall A \in \mathbf{A} \implies x \in \bigcap_{A \in \mathbf{A}} f^{-1}(A)\]

The proof is similar for union.

Theorem 2.34: Inverse Image of a Composition

Suppose \(f: X \rightarrow Y\) and \(g: Y \rightarrow W\) are functions. Then:

\[(g \circ f)^{-1} (A) = f^{-1} (g^{-1} (A))\]

for every \(A \subseteq W\)

Proof of Theorem 2.34:

Suppose \(A \subseteq W\), \(x \in X\), then we have:

\(x \in (g \circ f)^{-1} (A) \Longleftrightarrow g \circ f (x) \in A \Longleftrightarrow g(f(x)) \in A \Longleftrightarrow f(x) \in g^{-1} (A) \Longleftrightarrow x \in f^{-1}(g^{-1} (A))\)

Definition 2.35: Measurable Function

Suppose \((X, S)\) is a measurable space. A function \(f: X \rightarrow \mathbb{R}\) is called \(S\)-measurable (or just measurable if \(S\) is clear from the context)

if:

\[f^{-1}(B) \in S\]

for every Borel set \(B \in \mathbb{R}\).

Example 2.36: Measurable Functions

If \(S = \{\emptyset, X\}\), then the only \(S\)-measurable functions from \(X\) to \(\mathbb{R}\) are the constant functions, because otherwise each \(x \in X\) will have multiple \(f(x)\), which will violate the definition of function.

If \(S\) is the set of all subsets of \(X\), then every function from \(X \rightarrow \mathbb{R}\) is \(S\)-measurable.

If \(S = \{\emptyset, (-\infty, 0), [0, \infty), \mathbb{R}\}\), then a function \(f:\mathbb{R} \rightarrow \mathbb{R}\) is \(S\)-measurable IFF \(f\) is constant on \((-\infty, 0)\) and \(f\) is constant on \([0, \infty)\).

Definition 2.37: Characteristic Function; \(\chi_E\)

Suppose \(E\) is a subset of a set \(X\). The characteristic function of \(E\) is the function \(\chi_E: X \rightarrow \mathbb{R}\) defined by:

\[ \chi_E (x) = \begin{cases} 1, \quad \text{if} \; x \in E\\ 0, \quad \text{if} \; x \notin E \end{cases} \]

Theorem 2.39: Condition for Measurable Function

Suppose \((X, S)\) is a measurable space and \(f: X \rightarrow \mathbb{R}\) is a function such that:

\[f^{-1} ((a, \infty)) \in S\]

for all \(a \in \mathbb{R}\). Then \(f\) is an \(S\)-measurable function.

Where:

\[f^{-1}((a, \infty)) = \{x \in X: f(x) > a\}\]

We can replace the collection of sets \(\{(a, \infty): a\in \mathbb{R}\}\) by any collection of subsets of \(\mathbb{R}\) s.t the smallest \(\sigma\)-algebra containing that collection contains the Borel subset of \(\mathbb{R}\).

Definition 2.40: Borel Measurable Function

Suppose \(X \subseteq \mathbb{R}\). A function \(f: X \rightarrow \mathbb{R}\) is called Borel Measurable if \(f^{-1}(B)\) is a Borel set for every Borel set \(B \subseteq \mathbb{R}\).

If \(X\) is a Borel subset of \(\mathbb{R}\), then \(S\) might be the set of Borel sets contained in \(X\), in which case the phrase Borel Measurable is the same as \(S\)-measuable.

If \(X \subseteq \mathbb{R}\), then \(f\) is Borel measurable implies that \(X\) is a borel set.

If \(X \subseteq \mathbb{R}\), then \(f\) is Borel measurable IFF \(f^{-1}((a, \infty))\) is a borel set for all \(a \in \mathbb{R}\). Most of the proofs of Borel measurable function involves this result.

Definition 2.41: Every Continuous Function is Borel Measurable

Every continuous real-valued function defined on a Borel subset of \(\mathbb{R}\) is a Borel measurable function.

Definition 2.42: Increasing Function; Strictly Increasing

Suppose \(X \subseteq \mathbb{R}\) and \(f: X \rightarrow \mathbb{R}\) is a function:

• \(f\) is called increasing if \(f(x) \leq f(y)\) for all \(x, y \in X\) with \(x < y\).
• \(f\) is called strictly increasing if \(f(x) < f(y)\), for all \(x, y \in X\) with \(x < y\).

Theorem 2.43: Every Increasing Function is Borel Measurable

Every increasing function defined on a Borel subset is Borel measurable.

similar results for decreasing function.

Proof of Theorem 2.43:

Let \(X \subseteq \mathbb{R}\) be a borel set, let \(f: X \rightarrow \mathbb{R}\) be an increasing function. Then:

Let \(b = \inf f^{-1}((a, \infty))\). So we can write it as:

\[f^{-1}((a, \infty)) = [b, \infty) \cap X\]

Thus, we have \(f^{-1}((a, \infty))\) is a borel set which implies that \(f\) is borel measurable.

Theorem 2.44: Composition of Measurable Functions

Suppose \((X, S)\) is a measurable space and \(f: X \rightarrow \mathbb{R}\) is an \(S\)-measurable function. Suppose \(g\) is a real-valued Borel measurable function defined on a subset of \(\mathbb{R}\) that includes the range of \(f\). Then \(g \circ f: X \rightarrow \mathbb{R}\) is an \(S\)-measurable function.

Example 2.45

If \(f\) is measurable, then so are \(-f, |f|, f^2\).

Since \(g(x) = -x\), \(g(x) = |x|\), \(g(x) = x^2\) are all continuous functions defined on \(\mathbb{R}\), which is a borel subset of \(\mathbb{R}\), so \(g(f(x))\) is measurable.

Theorem 2.46: Algebraic Operations with Measurable Functions

Suppose \((X, S)\) is a measurable space and \(f, g: X \rightarrow \mathbb{R}\) are \(S\)-measurable. Then:

1. \(f + g\), \(f - g\), and \(fg\) are \(S-measurable\) functions.
2. If \(g(x) \neq 0\) for all \(x \in X\), then \(\frac{f}{g}\) is an \(S\)-measurable function.

Theorem 2.48: Limit of \(S\)-Measurable Functions

Suppose \((X, S)\) is a measurable space and \(f_1, f_2, ...\) is a sequence of \(S\)-measurable functions from \(X \rightarrow \mathbb{R}\). Suppose \(\lim_{k \rightarrow \infty} f_k (x)\) exists for each \(x \in X\). Define \(f: X \rightarrow \mathbb{R}\) by:

\[f(x) = \lim_{k \rightarrow \infty} f_k (x)\]

Then, \(f\) is an \(S\)-measurable function.

Definition 2.50: Borel Subsets of \([-\infty, \infty]\) (Extended definition of borel subset to \(\mathbb{R} \cup \{-\infty, \infty\}\))

A subset of \([-\infty, \infty]\) is called a Borel set if its intersection with \(\mathbb{R}\) is a Borel Set.

\([-\infty, \infty]\) is a borel set because \([-\infty, \infty] \cap \mathbb{R} = \mathbb{R}\) is a borel set.

Definition 2.51: Measurable Function

Suppose \((X, S)\) is a measurable space. A function \(f: X \rightarrow [-\infty, \infty]\) is called \(S\)-measurable if:

\[f^{-1}(B) \in S\]

Theorem 2.52: Condition for Measurable Function

Suppose \((X, S)\) is a measurable space and \(f: X \rightarrow [-\infty, \infty]\) is a function s.t:

\[f^{-1} ((a, \infty]) \in S\]

for all \(a \in \mathbb{R}\). Then \(f\) is an \(S\)-measurable function.

Theorem 2.53: Infimum and Supremum of a Sequence of \(S\)-measurable Functions

Suppose \((X, S)\) is a measurable space and \(f_1, f_2, ...\) is a sequence of \(S\)-measurable functions from \(X\) to \([-\infty, \infty]\). Define \(g, h: X \rightarrow [-\infty, \infty]\). Define \(g, h: X \rightarrow [-\infty, \infty]\) by:

\[g(x) = \inf\{f_k (x): k \in \mathbb{Z}^+\}\]

and

\[h(x) = \sup\{f_k (x): k \in \mathbb{Z}^+\}\]

Then \(g, h\) are \(S\)-measurable functions.

Proof of Theorem 2.53:

Suppose \(a \in \mathbb{R}\), The definition of the supremum implies that:

\[h^{-1} ((a, \infty]) = \{x; h(x) > a\}\]

Since,

\[\{x; h(x) > a\} \subseteq \bigcup_{k=1}^{\infty} \{x; f_k(x) > a\}\]

On the other hand, for all \(x \in \bigcup_{k=1}^{\infty} \{x; f_k(x) > a\}\), we have \(\sup_k f_k (x) > a\), thus:

\[\bigcup_{k=1}^{\infty} \{x; f_k(x) > a\}\subseteq \{x; h(x) > a\} \]

This implies that:

\[h^{-1} ((a, \infty]) = \{x; h(x) > a\} = \bigcup_{k=1}^{\infty} \{x; f_k(x) > a\} = \bigcup_{k=1}^{\infty} f^{-1}_k ((a, \infty])\]

Since \(f^{-1}_k ((a, \infty]) \in S\) by assumption, we have:

\[h^{-1} ((a, \infty]) \in S\]

Note that:

\[g(x) = - \sup\{-f_k (x): k \in \mathbb{Z}^+\}\]

for all \(x \in X\). The result about supremum implies that \(g\) is \(S\)-measurable.

Measures and Their Properties

The word measure allows us to use a single word instead of repeating theorems for length, area and volume.

Definition 2.54: Measure

Suppose \(X\) is a set and \(S\) is a \(\sigma\)-algebra on \(X\). A measure on \((X, S)\) is a function \(\mu: S \rightarrow [0, \infty]\) s.t \(\mu(\emptyset) = 0\) and:

\[\mu(\bigcup_{k=1}^{\infty} E_k) = \sum^{\infty}_{k=1} \mu(E_k)\]

for every disjoint sequence \(E_1, E_2, ...\) of sets in \(S\).

Notice that countable additivity implies finite additivity by constructing disjoint sequence \(E_1, E_2, ..., E_n, \emptyset, \emptyset, ...\)

Definition 2.56: Measure Space

A measure space is an ordered triple \((X, S, \mu)\), where \(X\) is a set, \(S\) is a \(\sigma\)-algebra on \(X\), and \(\mu\) is a measure on \((X, S)\).

Theorem 2.57: Measure Preserves Order; Measure of a Set Difference

Suppose \((X, S, \mu)\) is a measure space and \(D, E \in S\) are such that \(D \subseteq E\). Then:

1. \(\mu(D) \leq \mu(E)\)
2. \(\mu(E / D) = \mu(E) - \mu(D)\) provided that \(\mu(D) < \infty\)

Proof of Theorem 2.57:

Suppose \(D \subseteq E\), then \(E = D \cup (E / D)\) and these two sets are disjoint, that is:

\[\mu(E) = \mu(D \cup (E / D)) = \mu(D) + \mu(E / D) \geq \mu(D)\]

given that \(\mu(D) < \infty\) subtract both sides by \(\mu(D)\), we have b.

Theorem 2.58: Countable Subadditivity

Suppose \((X, S, \mu)\) is a measure space and \(E_1, E_2, ... \in S\). Then:

\[\mu(\bigcup^{\infty}_{k=1} E_k) \leq \sum^{\infty}_{k=1}\mu(E_k)\]

Theorem 2.59: Measure of an Increasing Union

Suppose \((X, S, \mu)\) is a measure space and \(E_1 \subseteq E_2 \subseteq ...\) is an increasing sequence of sets in \(S\). Then:

\[\mu(\bigcup^\infty_{k=1} E_k) = \lim_{k\rightarrow \infty} \mu(E_k)\]

Proof Theorem 2.59

If \(\mu(E_k) = \infty\) for some \(k \in \mathbb{Z}^+\), then the equation above holds because both sides equals \(\infty\):

\[\mu(\bigcup^{\infty}_{k=1} E_k) \geq \mu(E_k) = \infty = \lim_{k\rightarrow \infty} \mu(E_k)\]

Thus, we consider only the case where \(\mu(E_k) < \infty\) for all \(k \in \mathbb{Z}^+\).

Let \(E_0 = \emptyset\). Then:

\[\bigcup^{\infty}_{k=1} E_k = E_1 \cup (E_2 / E_1) \cup (E_3 / E_2) \cup ... = \bigcup^{\infty}_{j=1} E_{j} / E_{j-1}\]

And the union on the right hand side is disjoint, that is:

\[\mu(\bigcup^{\infty}_{j=1} E_{j} / E_{j-1}) = \sum^{\infty}_{j=1} \mu(E_{j} / E_{j-1}) = \lim_{k\rightarrow \infty} \sum^{k}_{j=1} \mu(E_{j} / E_{j-1})\]

Since \(E_{j-1} < \infty\), we have:

\[\lim_{k\rightarrow \infty} \sum^{k}_{j=1} \mu(E_{j} / E_{j-1}) = \lim_{k\rightarrow \infty} \sum^{k}_{j=1} \mu(E_{j}) - \mu(E_{j-1}) = \lim_{k\rightarrow \infty} \mu(E_k)\]

Theorem 2.60: Measure of a Decreasing Intersection

Suppose \((X, S, \mu)\) is a measure space and \(E_1 \supseteq E_2 \supseteq ...\) is a decreasing sequence of sets in \(S\), with \(\mu(E_1) < \infty\). Then:

\[\mu(\bigcap^{\infty}_{k=1}E_k) = \lim_{k\rightarrow \infty} \mu(E_k)\]

Theorem 2.61: Measure of a Union

Suppose \((X, S, \mu)\) is a measure space and \(D, E \in S\), with \(\mu(D \cap E) < \infty\). Then:

\[\mu(D \cup E) = \mu(D) + \mu(E) - \mu(D \cap E)\]

Lebesgue Measure

The main goal of this section is to prove that outer measure when restrict to borel sets is a measure.

Theorem 2.62: Additivity of Outer Measure If One of the Sets is Open

Suppose \(A, G\) are disjoint subsets of \(\mathbb{R}\) and \(G\) is open. Then:

\[|A \cup G| = |A| + |G|\]

Theorem 2.63: Additivity of Outer Measure IFF the Sets is Closed

Suppose \(A, F\) are disjoint subsets of \(\mathbb{R}\) and \(\mathbb{F}\) is closed. Then:

\[|A \cup F| = |A| + |F|\]

Theorem 2.65: Approximation of Borel Sets from Below by Closed Sets

Suppose \(B \subseteq \mathbb{R}\) is a Borel set. Then for every \(\epsilon > 0\), there exists a closed set \(F \subseteq B\) s.t \(|B / F| < \epsilon\).

Theorem 2.66: Additivity of Outer Measure If One of the Sets is a Borel Set

Suppose \(A\) and \(B\) are disjoint subsets of \(\mathbb{R}\) and \(B\) is a Borel set. Then:

\[|A \cup B| = |A| + |B|\]

Theorem 2.67: Existence of a Subset of \(\mathbb{R}\) that is not a Borel Set

There exists a set \(B \subseteq \mathbb{R}\) s.t \(|B| < \infty\) and \(B\) is not a Borel set.

Proof of Theorem 2.67:

From Theorem 2.18, we know that there exists disjoint set \(A, B \subseteq \mathbb{R}\) s.t:

\[|A \cup B| \neq |A| + |B|\]

Then, according to Theorem 2.66, if all subsets of \(\mathbb{R}\) are Borel sets, we would have violated Theorem 2.18.

Theorem 2.68: Outer Measure is a Measure on Borel Sets

Outer measure is a measure on \((\mathbb{R}, \mathbb{B})\), where \(\mathbb{B}\) is the \(\sigma\)-algebra of Borel subsets of \(\mathbb{R}\).

Definition 2.69: Lebesgue Measure (on \(\mathbb{B}\))

Lebesgue Measure is the measure on \((\mathbb{R}, \mathbb{B})\), where \(\mathbb{B}\) is the \(\sigma\)-algebra of Borel subsets of \(\mathbb{R}\), that assigns to each Borel set its outer measure.

In other words, the Lebesgue measure of a set is the same as its outer measure except it should not be applied to arbitrary sets but only to Borel sets.

Lebesgue Measurable Sets

Definition 2.70: Lebesgue Meaurable Set

A set \(A \subseteq \mathbb{R}\) is called Lebesgue measurable if there exists a Borel set \(B \subseteq A\) s.t \(|A / B| = 0\).

This definition implies that all Borel set is Lebesgue measurable because if \(A \subseteq \mathbb{R}\) is a Borel set, then we can take \(B = A\). If \(A\) is a set with outer measure \(0\), then \(A\) is Lebesgue measurable because we can take \(B = \emptyset\).

Theorem 2.71: Equivalences for Being a Lebesgue Measurable Set

Suppose \(A \subseteq \mathbb{R}\). Then the following are equivalent:

1. \(A\) is Lebesgue measurable.
2. For each \(\epsilon > 0\), there exists a closed set \(F \subseteq A\) with \(|A / F| \leq \epsilon\).
3. There exists closed sets \(F_1, F_2, ...\) contained in \(A\) s.t \(|A / \bigcup^{\infty}_{k=1} F_k| = 0\).
4. There exists a Borel set \(B \subseteq A\) s.t \(|A / B| = 0\).
5. For each \(\epsilon > 0\). There exists an open set \(G \supseteq A\) s.t \(|G / A| < \epsilon\).
6. There exists open sets \(G_1, G_2, ...\) containing \(A\) s.t \(|(\bigcap^{\infty}_{k=1})G_k / A| = 0\).
7. There exists a Borel set \(B \subseteq A\) s.t \(|B / A| = 0\).

Theorem 2.72: Outer Measure is a Measure on Lebesgue Measurable Sets

1. The set \(L\) of Lebesgue measurable subsets of \(\mathbb{R}\) is a \(\sigma\)-algebra on \(\mathbb{R}\).
2. Outer measure is a measure on \((\mathbb{R}, L)\).

b implies that there are sets in \(\mathbb{R}\) that is not Lebesgue measurable.

Proof of Theorem 2.72:

We know that from theorem 2.71 that a and b are equivalent, that is, let \(L\) be the set that contains all \(A\) that satisfies b, we can show that \(L\) is \(\sigma\)-algebra.

To show b, let \(A_1, A_2, ... \in L\) be a disjoint sequence of Lebesgue measurable sets, then by definition, we have Borel set \(B_k \subseteq A_k\) s.t

\[|A_k / B_k| = 0\]

for all \(k\). Then, \(B_1, B_2, ..\) is a sequence of disjoint Borel sets.

Then we have:

\[|\bigcup^{\infty}_{k=1} A_k| \geq |\bigcup^{\infty}_{k=1} B_k| = \sum^{\infty}_{k=1} |B_k|\]

Since \(|A_k| = |B_k \cup (A_k / B_k)| \leq |B_k| + |A_k / B_k| = |B_k|\), we have:

\[|\bigcup^{\infty}_{k=1} A_k| \geq \sum^{\infty}_{k=1} |B_k| \geq \sum^{\infty}_{k=1} |A_k|\]

Definition 2.73: Lebesgue Measure (More general, on \(L\))

Lebesgue measure is the measure on \((\mathbb{R}, L)\), where \(L\) is the \(\sigma\)-algebra of Lebesgue measurable subsets of \(\mathbb{R}\), that assigns to each Lebesgue measurable set its outer measure.

Every Lebesgue measurable set differs from a Borel set by a set with outer measure \(0\).

The two definitions differ only on the domain, which will be specified unless it is irrelevant.

Definition 2.74: Cantor Set

The Cantor set \(C\) is \([0, 1] / (\bigcup^{\infty}_{n=1}) G_n\), where \(G_1 = (\frac{1}{3}, \frac{2}{3})\) and \(G_n\) for \(n > 1\) is the union of the middle-third open intervals in the intervals of \([0, 1] / \bigcup^{n-1}_{j=1}G_j\).

One way to envision the Cantor set \(C\) is to start with the interval \([0, 1]\) and then consdier the process that removes at each step the middle-third open intervals of all intervals left from the previous step.

Theorem 2.76: \(C\) is Closed, has Measure \(0\), and Contains no nontrivial Intervals

1. The Cantor set is a closed subset of \(\mathbb{R}\).
2. The Cantor set has Lebesgue measure \(0\).
3. The Cantor set contains no interval with more than one element.

Convergence of Measurable Functions

Definition 2.82: Pointwise and Uniform Convergence

Suppose \(X\) is a set, \(f_1, f_2, ...\) is a sequence of functions from \(X \rightarrow \mathbb{R}\), and \(f\) is a function from \(X \rightarrow \mathbb{R}\).

• The sequence \(f_1, f_2, ...\) converges pointwise on \(X\) to \(f\), if:

  \[\lim_{k \rightarrow \infty} f_k (x) = f(x)\]

  for each \(x \in X\). In other words, \(f_1, f_2, ...\) converges pointwise on \(X\) to \(f\) if for each \(x \in X\) and every \(\epsilon > 0\), there exists a \(n > \mathbb{Z}^+\) s.t \[|f_k(x) - f(x)| < \epsilon\] for all integers \(k \geq n\).

• The sequence \(f_1, f_2, ...\) converges uniformly on \(X\) to \(f\) if for every \(\epsilon > 0\), there exists \(n \in \mathbb{Z}^+\) s.t \(|f_k (x) - f(x)| < \epsilon\), forall integers \(k \geq n\) and for all \(x \in X\).

Theorem 2.84: Uniform Limit of Continuous Function is Continuous

Suppose \(B \subseteq \mathbb{R}\) and \(f_1, f_2, ...\) is a sequence of functions from \(B \rightarrow \mathbb{R}\) that converges uniformly on \(B\) to a function \(f: B \rightarrow \mathbb{R}\). Suppose \(b \in B\) and \(f_k\) is continuous at \(b\) for each \(k \in \mathbb{Z}^+\). Then \(f\) is continuous at \(b\).

Theorem 2.85: Egorov's Theorem

Suppose \((X, S, \mu)\) is a measure space with \(\mu(X) < \infty\). Suppose \(f_1, f_2, ...\) is a sequence of \(S\)-measurable functions from \(X \rightarrow \mathbb{R}\) that converges pointwise on \(X\) to a function \(f: X \rightarrow \mathbb{R}\). Then for every \(\epsilon > 0\), there exists a set \(E \subseteq S\) s.t \(\mu(X / E) < \epsilon\) and \(f_1, f_2, ...\) converges uniformly to \(f\) on \(E\).

Proof of Theorem 2.85:

Suppose \((X, S, \mu)\) is a measure space with \(\mu(X) < \infty\). Suppose \(f_1, f_2, ...\) is a sequence of \(S\)-measurable functions from \(X \rightarrow \mathbb{R}\) that converges pointwise on \(X\) to a function \(f: X \rightarrow \mathbb{R}\).

We want to prove that:

1. \(E = \bigcap^{\infty}_{n=1} A_{m_n, n} \in S\).
2. \(\mu(X / E) < \epsilon\).
3. \(f_1, f_2, ...\) converges uniformly to \(f\) on \(E\).

Suppose \(\epsilon > 0\), then for any \(n \in \mathbb{Z}^+\), the definition of pointwise convergence implies that for some \(k\) for some \(x\):

\[|f_k (x) - f(x)| < \frac{1}{n}\]

then:

\[\bigcap^{\infty}_{k=1}\{x \in X: |f_k (x) - f(x)| < \frac{1}{n}\} = X\]

\[\bigcup^{\infty}_{m=1}\bigcap^{\infty}_{k=m}\{x \in X: |f_k (x) - f(x)| < \frac{1}{n}\} = X\]

For \(m \in \mathbb{Z}^+\), let:

\[A_{m, n} = \bigcap^{\infty}_{k=m}\{x \in X: |f_k (x) - f(x)| < \frac{1}{n}\}\]

Since \(f_1, f_2, ...\) is \(S\)-measurable, by theorem 2.48, \(f\) is \(S\)-measurable, by theorem 2.46 \(f_k - f\) is \(S\)-measurable, this implies that:

\[(f_k - f)^{-1} ((-\frac{1}{n}, \frac{1}{n})) \in S\]

So \(A_{m, n} \in S\) which proves 1.

We can clearly see that \(A_{1, n} \subseteq A_{2, n} \subseteq ...\) is an increasing sequence of sets, and by theorem 2.59:

\[\lim_{m \rightarrow \infty} \mu(\bigcup^{\infty}_{m=1} A_{m, n}) = \mu(X)\]

Thus, by sequence convergence theorem, there exists \(m_n \in \mathbb{Z}^+\) s.t:

\[\mu(X) - \mu(A_{m_n, n}) < \frac{\epsilon}{2^n}\]

Let \(E = \bigcap^{\infty}_{n=1} A_{m_n, n}\), then:

\[\mu(X / E) = \mu(X / \bigcap^{\infty}_{n=1} A_{m_n, n}) = \mu(\bigcup^{\infty}_{n=1}(X / A_{m_n, n})) \leq \sum^{\infty}_{n=1}\mu(X / A_{m_n, n}) = \sum^{\infty}_{n=1}\mu(X) - \mu(A_{m_n, n})\]

Since \(\sum^{\infty}_{n=1} \epsilon (\frac{1}{2})^n = (\sum^{\infty}_{n=0} \epsilon (\frac{1}{2})^n) - \epsilon = \epsilon\), we have:

\[\mu(X / E) < \epsilon\]

which proves 2.

Now, suppose \(\epsilon^\prime > 0\) and let \(n\) be such that \(\frac{1}{n} < \epsilon^\prime\). Then \(E \subseteq A_{m_n, n}\), which implies that:

\[|f_k (x) - f(x)| < \frac{1}{n} < \epsilon^\prime\]

for all \(k \geq m_n\) and \(x \in E\), thus, we have proved 3.

Definition 2.88: Simple Function

A function is called simple if it takes only finite many values.

Let \(f = c_1\chi_{E_1} + .... + c_n \chi_{E_n}\), where \(E_k = f^{-1}(\{c_k\})\), then \(f\) is \(S\)-measurable IFF \(E_1, ..., E_n \in S\)

The representation of the simple function \(f\) is not unique. By requiring \(c_1, ..., c_n\) to be distinct and \(E_1, ...., E_n\) to be nonempty and disjoint with \(E_1 \cup .... \cup E_n = X\) produces what is called the standard representation of a simple function (taking \(E_k = f^{-1} (\{c_k\})\) where \(c_1, c_2, ..., c_n\) are distinct values of \(f\))

Theorem 2.89: Approximation by Simple Function

Suppose \((X, S)\) is a measurable space and \(f: X \rightarrow [-\infty, \infty]\) is \(S\)-measurable. Then there exists a sequence \(f_1, f_2, ...\) of functions from \(X \rightarrow \mathbb{R}\) s.t:

1. Each \(f_k\) is simple \(S\)-measurable function.
2. \(|f_k(x)| \leq |f_{k+1}(x)| \leq |f(x)|\) for all \(k \in \mathbb{Z}^+\) and all \(x \in X\).
3. \(\lim_{k \rightarrow \infty} f_k (x) = f(x)\) for every \(x \in X\).
4. \(f_1, f_2, ...\) converges uniformly on \(X\) to \(f\) if \(f\) is bounded.

Theorem 2.91: Luzin's Theorem

Suppose \(g: \mathbb{R} \rightarrow \mathbb{R}\) is a Borel measurable function. Then for every \(\epsilon > 0\), there exists a closed set \(F \subseteq \mathbb{R}\) s.t \(|\mathbb{R} / F| < \epsilon\) and \(g|_{F}\) is a continuous function on \(F\).

In other words, if \(g\) is Borel measurable function, then there exists a closed set \(F \subseteq \mathbb{R}\), s.t the outer measure of the complement of \(F\) is arbitrarily large and \(g|_F\) is continuous on this arbitrarily small open set. \(g|_F\) is continuous on \(F\) is not the same as \(g\) is continuous at every point of \(B\).

\(\chi_\mathbb{Q}\) is discontinuous as every point of \(\mathbb{R}\), however, \(\chi_{\mathbb{Q}} |_{\mathbb{R} / \mathbb{Q}}\) is continuous everywhere on \(\mathbb{R} / \mathbb{Q}\) because they are all 0.

Theorem 2.92: Continuous Extensions of Continuous Functions

• Every continuous function on a closed subset of \(\mathbb{R}\) can be extended to a continuous function on all of \(\mathbb{R}\).
• More precisely, if \(F \in \mathbb{R}\) is closed and \(g: F \rightarrow \mathbb{R}\) is continuous, then there exists a continuous function \(h: \mathbb{R} \rightarrow \mathbb{R}\) s.t \(h|_F = g\).

Theorem 2.93: Luzin's Theorem, Second Version

Suppose \(E \subseteq \mathbb{R}\) and \(g: E \rightarrow \mathbb{R}\) is a Borel measurable function. Then for every \(\epsilon > 0\), there exists a closed set \(F \subseteq E\) and a continuous function \(h: \mathbb{R} \rightarrow \mathbb{R}\) s.t \(|E / F| < \epsilon\) and $h|_{F} = g |_F $.

Definition 2.94: Lebesgue Measurable Function

A function \(f: A \rightarrow \mathbb{R}\), where \(A \subseteq \mathbb{R}\), is called Lebesgue measurable if \(f^{-1}(B)\) is a Lebesgue measurable set for every Borel set \(B \subseteq \mathbb{R}\).

The definition implies that if \(A\) is Lebesgue measurable then \(A\) is a Lebesgue measurable subset of \(\mathbb{R}\). If \(A\) is a Lebesgue measurable subset of \(\mathbb{R}\) and \(S\) is a \(\sigma\)-algebra of all Lebesgue measurable subsets of \(A\), then \(f\) is \(S\)-measurable.

Theorem 2.95 Every Lebesgue Measurable Function is Almost Borel Measurable:

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a Lebesgue measurable function. Then there exists a Borel measurable function \(g: \mathbb{R} \rightarrow \mathbb{R}\) s.t:

\[|\{x \in \mathbb{R}: g(x) \neq f(x)\}| = 0\]

Review 2.96

• A Borel set is an element of the smallest \(\sigma\)-algebra on \(\mathbb{R}\) that contains all the open subsets of \(\mathbb{R}\).
• A Lebesgue measurable set is an element of the smallest \(\sigma\)-algebra on \(\mathbb{R}\) that contains all the open subsets of \(\mathbb{R}\) and all the subsets of \(\mathbb{R}\) with outer measure \(0\).
• Every Lebesgue measurable set differs from a Borel set by a set with outer measure \(0\).
• Outer measure restricted to the \(\sigma\)-algebra of Borel sets or Lebesgue measurable sets is called Lebesgue measure.
• A function \(f: A \rightarrow \mathbb{R}\) is called Borel measurable if \(f^{-1}(B)\) is a Borel set for every Borel set \(B \in \mathbb{B}\).
• A function \(f: A \rightarrow \mathbb{R}\) is called Lebesgue measurable if \(f^{-1}(B)\) is a Lebesgue measurable set for every Borel set \(B \in \mathbb{B}\).