0x022 Real Analysis

Measures

Outer Measure

Definition (length of open interval) The length $l(I)$ of an open interval $I$ is defined by

$$ l(I) =
\begin{cases}
b-a & \text{for } I = (a,b) \\
0 & \text{if } I = \emptyset \\ \infty & \text{if } I = (-\infty, a) \text{ or } I = (a, \infty) \\ \infty & \text{if } I = (-\infty, \infty) \end{cases} $$

Definition (outer measure, $|A|$) The outer measure $|A|$ of a set $A \subset R$ is defined by

$$ |A| = \inf \{ \sum_{k=1}^{\infty} l(I_k): I_1, I_2, … \text{ are open intervals and covers } A \}$$

Properties of Outer Measure

Property (countable set has measure 0) Every countable set of $R$ has outer measure 0

Property (outer measure preserves order)

$$A \subset B \implies |A| \leq |B|$$

Property (outer measure is translation invariant) Suppose $t \in R, A \subset R$

$$|t+A| = |A|$$

Property (countable subadditivity) Suppose $A_1, A_2, …$ is a sequence of subsets of $R$. Then

$$ |\bigcup_{k=1}^{\infty} A_k | \leq \sum_{k=1}^{\infty} |A_k|$$

Property (outer measure of a closed interval) Suppose $a, b \in R, a < b$, then

$$|[ a, b ] | = b-a$$

note that this can be proved with Heine-Borel

Property (nonadditivity of outer measure) There exists disjoint subsets $A, B \subset R$ such that

$$|A \cup B| \neq |A| + |B|$$

note that this can be proved using AoC

Measurable Spaces and Functions

Definition ($\sigma$-algebra) $X$ is a set and $S$ is a set of subsets of $S$, $S$ is called $\sigma$-algebra when the following conditions are satisfied:

  • $\emptyset \in S$
  • $E \in S \implies X \setminus E \in S$
  • $E_1, E_2, … \in S \implies \bigcup_{k=1}^{\infty} E_i \in S$

Definition (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.

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

Definition (inverse image) If $f: X \to Y$ is a function and $A \subset Y$, then the set $f^{-1}(A)$ is defined by

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

Definition (measurable function) Suppose $(X,S)$ is a measurable space. A function $f: X \to R$ is called S-measurable function if for every Borel set $B \subset R$

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

Criterion Suppose $(X,S)$ is a measurable space and $f: X \to R$ is a measurable function iff

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

Definition (Borel measurable function) Suppose $X \subset R$. A function $X \to R$ is called Borel measurable if $f^{-1}(B)$ is a Borel set for every Borel set $B \subset R$

Example Every continuous real-valued function defined on a Borel subset is a Borel measurable function

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

Criterion (operations preserving measurable functions) Measurable functions are closed under composition, algebraic operations and pointwise functional limit

Measures and Their Properties

Definition (measure) Suppose $X$ is a set and $S$ is a $\sigma$-algebra on $X$. A measure on $(X,S)$ is a function $\mu: S \to [0, \infty]$ such that $\mu(\emptyset) = 0$ and

$$\mu(\bigcup_{k=1}^{\infty} E_k) = \sum_{k=1}^{\infty} \mu(E_k)$$

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

Property (measure preserves order) Suppose $(X, S, \mu)$ is a measure space, $D, E \in S$ and $D \subset E$, then
$$\mu(D) \leq \mu(E)$$

Property (acountable subadditivity) Suppose $(X, S, \mu)$ is a measure space and $E_1, E_2, … \in S$, then

$$\mu(\cup_{k=1}^{\infty} E_i) \leq \sum_{k=1}^{\infty} \mu(E_i)$$

Property (measure of a union) Suppose $(X, S, \mu)$ is a measure space and $D,E \in S, \mu(D \cap E) < \infty $, then

$$\mu (D \cup E) = \mu(D) + \mu(E) – \mu(D \cap E)$$

Property (outer measure is a measure on Borel sets) Outer measure is a measure on $(R,B)$ where $B$ is the $\sigma$-algebra of Borel subsets of $R$

Definition (Lebesgue measure wrt Borel subsets) Lebesgue measure is the measure on $(R,B)$ where $B$ is the $\sigma$ algebra of Borel subsets of $R$, that assigns to each Borel set its outer measure

Definition (Lebesgue Measurable set) A set $A \subset R$ is called Lebesgue Measurable if there exists a Borel set $B \subset A$ such that $| A \ B|=0$

Definition (Lebesgue Measure wrt Lebesgue Measurable set) Lebesgue Measure is the measure on $(R, L)$ where $L$ is the $\sigma$-algebra of Lebesgue measurable subsets of $R$, that assigns to each Lebesgue measurable set its outer measure. $L$ is defined as

$$ L = \{ D \subset R | \forall{\epsilon > 0} \exists{F \subset D} |D \ F | < \epsilon \}$$

Convergence of Measurable Functions

Theorem (Egorov) Suppose $(X,S,\mu)$ is a measurable space and $\mu(X) < \infty$. Suppose $f_1, f_2, …$ is a sequence of $S$-measurable function from $X$ to $R$ that converges pointwise on $X$ to a function $f$. Then for every $\epsilon > 0$, there exists a set $E \in S$ such that $\mu(X \ E ) < \epsilon$ and $f_1, f_2…$ converges uniformly to $f$ on $E$

Theorem (Luzin) Suppose $g: R \to R$ is a Borel measurable function. Then for every $\epsilon > 0$ there exists a closed set $F \subset R$ such that $| R \ F| < \epsilon$ and $g|_{F}$ is continuous function on $F$

Integration

For a measure space $(X, S, \mu)$, we aims at defining $\int_{A} f(x) d\mu$ for suitable measurable function $f: X \to R$ and for any $A \in S$ in a systematic manner.

Integration with respect to a Measure

Definition (S-partition) Suppose $S$ is a $\sigma$-algebra on a set $X$. An $S$-partition of $X$ is a finite collection $A_1, …, A_m$ of disjoint sets in $S$ such that $A_i \cup …. \cup A_m = X$

Definition (lower Lebesgue sum) Suppose $(X, S, \mu)$ is a measure space, $f: X \to [0, \infty]$ is a S-measurable function and $P$ is a partition $A_1, …, A_m$ of $X$. The lower Lebesgue sum $\mathcal{L}(f,P)$ is defined by

$$\mathcal{L}(f, P) = \sum_{j=1}^{m} \mu(A_j) \inf_{A_j} f$$

We first think about reducing integration to the simple case of integrating non-negative function $f^{+}$, later it can be extended to a general function $f$ by expressing it using two non-negative function $f = f^{+} – f^{-}$

Definition (Integral of a nonnegative function) Suppose $(X, S, \mu)$ is a measure space and $f: X \to [0, \infty]$ is an $S$-measurable function. The integral of $f$ with respect to $\mu$ is defined by

$$ \int f d\mu = \sup \{ \mathcal{L}(f, P) | P \text{ is an S-partition of }X \}$$

Next, we think about reducing it to the case of integrating simple functions.

Lemma (Integral of a characteristic function) Suppose $(X, S, \mu)$ is a measure space and $E \in S$, Then

$$\int \chi_{E} d\mu = \mu(E)$$

Theorem (monotone convergence) Suppose $(X, S, \mu)$ is a measure space and $0 \leq f_1 \leq f_2 \leq …$ is an increasing sequence of $S$-measurable functions. Define $f: X \to [0, \infty]$ by

$$f(x) = \lim_{k \to \infty} f_k(x)$$

Then

$$\lim_{k \to \infty} \int f_k(x) d\mu = \int f(x) d\mu$$

Theorem (addivity of integration) Suppose $(X, S, \mu)$ is a measure space and $f, g: X \to R$ are S-measurable functions such that $\int |f| < \infty, \int |g| < \infty$. Then

$$\int (f+g) d\mu = \int f d\mu + \int g d\mu$$

Theorem (integration is homogeneous) Suppose $(X, S, \mu)$ is a measure space and $f: X \to [-\infty, \infty]$ is a function such that $\int f d\mu$ is defined. If $c \in R$, then

$$\int c f d\mu = c \int f d\mu$$

Limits of Integrals

Definition (integration on a subset) Suppose $(X, S, \mu)$ is a measure space and $E \in S$. If $f: X \to [-\infty, \infty]$ is an $S$-measurable function, then $\int_{E} f d\mu$ is defined by

$$\int_{E} f d\mu = \int \chi_{E} f d\mu$$

Definition (almost every) Suppose $(X, S, \mu)$ is a measure space. A set $E \in S$ is said to contain $\mu$-almost every element of $X$ if $\mu( X \ E)=0$.

Theorem (Dominated Convergence Theorem) Suppose $(X, S, \mu)$ is a measure space, $f: X \to [-\infty, \infty]$ is $S$-measurable, and $f_1, f_2, …$ are $S$-measurable functions from $X$ to $[-\infty, \infty]$ such that

$$\lim_{k \to \infty} f_k (x) = f(x)$$

for almost every $x \in X$. If there exists an $S$-measurable function $g: X \to [0, \infty]$ such that

$$\int g d\mu < \infty \land | f_k (x) | \leq g(x) $$

for every $k \in Z^+$ and almost every $x \in X$, then

$$lim_{k \to \infty} \int f_k(x) d\mu = \int f d\mu$$

Definition ($||f||_1, \mathcal{L}^1(\mu)$) Suppose $(X, S, \mu)$ is a measure space. If $f: X \to [-\infty, \infty]$ is $S$-measurable. then the $\mathcal{L}^1$-norm of $f$ is defined by

$$||f||_1 = \int |f| d\mu$$

The Lebesgue space $\mathcal{L}^1(\mu)$ is defined by

$$\mathcal{L}^1(\mu) = \{ f: f \text{ is an S-measurable function from X to R such that } ||f||_1 < \infty$$

Differentiation

Product Measures

Products of Measure Spaces

Definition (product of $\sigma$-algebra, $S \otimes T$) Suppose $(X, S), (Y, T)$ are measurable spaces. Then the product $S \otimes T$ is defined to be the smallest $\sigma$-algebra on $X \times Y$ contains

$$\{ A \times B | A \in S, B \in T \}$$

Definition (cross sections of sets) Suppose $X, Y$ are sets and $E \subset X \times Y$. Then for $a \in X, b \in Y$, the cross sections $[E]_a, [E]^b$ are defined by

$$[E]_a = \{ y \in Y | (a, y) \in E \}$$

$$[E]^b = \{ x \in X | (x, b) \in E \}$$

Banach Space

Theorem (Baire) A complete metric space is not the countable union of closed subsets with empty interior

LP Space

Definition ($||f||_p$) Suppose that $(X, S, \mu)$ is a measure space, $0 < p < \infty$ and $f: X \to F$ is $S$-measurable. Then the p-norm of $f$ is defined by

$$||f||_{p} = \Big( \int |f|^p d\mu \Big)^{1/p}$$

Note that the exponent is to make sure $||af||_p = |a| || f||_p$

Definition (essential supremum) The essential supremum of $f$ is

$$||f||_{\infty} = \inf \{ t > 0 | \mu(\{ x \in X | |f(x)| > t \} ) = 0 \}$$

Hilbert Space

Reference

[1] Axler, Sheldon. “Measure, Integration & Real Analysis.” (2020): 411.