Measure Theory Formula Sheet -

σ-algebras

A collection $\mathcal{F}$ of subsets of $X$ is a σ-algebra if:

$\emptyset \in \mathcal{F}$
If $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$
If $A_n \in \mathcal{F}$ for $n \in \mathbb{N}$, then $\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}$

Generated σ-algebra: $\sigma(\mathcal{C})$ = smallest σ-algebra containing $\mathcal{C}$

Closure under countable intersections: $\bigcap_{n=1}^{\infty} A_n \in \mathcal{F}$

Borel σ-algebra

On $\mathbb{R}^n$: $\mathcal{B}(\mathbb{R}^n) = \sigma(\text{open sets})$

Also generated by: $(a,b)$, $[a,b]$, $(a,b]$, $[a,b)$

Borel sets include: intervals, countable unions, countable intersections, complements

$\mathbb{Q} \in \mathcal{B}(\mathbb{R})$ (countable union of singletons)

Measures

A measure on $(X, \mathcal{F})$ is $\mu: \mathcal{F} \to [0,\infty]$ with:

$\mu(\emptyset) = 0$
Countable additivity: $\mu(\bigcup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} \mu(A_n)$ (disjoint)

Measure space: $(X, \mathcal{F}, \mu)$

Measure Properties

Monotonicity: $A \subseteq B \Rightarrow \mu(A) \leq \mu(B)$

Subadditivity: $\mu(\bigcup A_n) \leq \sum \mu(A_n)$

Continuity from below: $A_n \uparrow A \Rightarrow \mu(A_n) \to \mu(A)$

Continuity from above: $A_n \downarrow A \Rightarrow \mu(A_n) \to \mu(A)$ (if $\mu(A_1) < \infty$)

Outer Measures

$\mu^*: \mathcal{P}(X) \to [0,\infty]$ with:

$\mu^*(\emptyset) = 0$
Monotone: $A \subseteq B \Rightarrow \mu^*(A) \leq \mu^*(B)$
Countable subadditivity: $\mu^*(\bigcup A_n) \leq \sum \mu^*(A_n)$

Carathéodory: $E$ is measurable iff for every test set $T$, $\mu^*(T) = \mu^*(T \cap E) + \mu^*(T \cap E^c)$

Lebesgue Measure on $\mathbb{R}^n$

$\lambda^n((a_1,b_1) \times \cdots \times (a_n,b_n)) = \prod_{i=1}^n (b_i - a_i)$

Lebesgue measure on $\mathbb{R}$: $\lambda(I) = $ length of interval $I$

$\lambda(\mathbb{Q}) = 0$ (countable sets have measure zero)

Translation invariant: $\lambda(A + x) = \lambda(A)$

Measurable Functions

$f: X \to \mathbb{R}$ is measurable if $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(\mathbb{R})$

Equivalent: $\{x: f(x) > a\} \in \mathcal{F}$ for all $a \in \mathbb{R}$

Continuous functions are measurable. Products, sums, limits of measurable functions are measurable

Simple Functions

$s = \sum_{i=1}^n a_i \mathbf{1}_{A_i}$ where $A_i$ are disjoint measurable sets, $a_i \in \mathbb{R}$

Every non-negative measurable function is limit of simple functions: $f_n \uparrow f$

Simple functions approximate measurable functions monotonically

Lebesgue Integral

For simple $s = \sum a_i \mathbf{1}_{A_i}$: $\int s \, d\mu = \sum a_i \mu(A_i)$

For non-negative measurable $f$: $\int f \, d\mu = \sup\{\int s \, d\mu : 0 \leq s \leq f, s \text{ simple}\}$

General $f$: $f = f^+ - f^-$, integrate if at least one integral is finite

Integral Properties

Linearity: $\int (af + bg) \, d\mu = a\int f \, d\mu + b\int g \, d\mu$

Monotonicity: $f \leq g \Rightarrow \int f \, d\mu \leq \int g \, d\mu$

$\int_A f \, d\mu = \int f \mathbf{1}_A \, d\mu$

If $\mu(E) = 0$, then $\int_E f \, d\mu = 0$

Monotone Convergence Theorem

If $0 \leq f_n \uparrow f$ (pointwise a.e.), then: $$\int f_n \, d\mu \to \int f \, d\mu$$

Justifies interchange of limit and integral for monotone sequences

Also: $\int \sum_{n=1}^{\infty} f_n = \sum_{n=1}^{\infty} \int f_n$ (non-negative $f_n$)

Fatou's Lemma

If $f_n \geq 0$, then: $$\int \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int f_n \, d\mu$$

Lower semicontinuity of integral w.r.t. convergence

Converse: $\limsup \int f_n \leq \int \limsup f_n$ if $|f_n| \leq g$ and $\int g < \infty$

Dominated Convergence Theorem

If $f_n \to f$ a.e., $|f_n| \leq g$ with $\int g < \infty$, then: $$\int f_n \, d\mu \to \int f \, d\mu$$

Also: $\int |f_n - f| \, d\mu \to 0$

Most powerful: requires pointwise convergence + domination

Fubini's Theorem

For $f \geq 0$ on $(X \times Y, \mathcal{F} \otimes \mathcal{G})$, with $\mu,\nu$ σ-finite (this is Tonelli's theorem): $$\int f \, d(\mu \times \nu) = \int \left(\int f(x,y) \, d\mu(x)\right) d\nu(y)$$

Fubini-Tonelli: Iterated integrals for non-negative functions need not assume integrability

Radon-Nikodym Theorem

If $\nu \ll \mu$ (absolutely continuous) and $\mu$ is σ-finite, then $\exists!$ $f \geq 0$ a.e. with: $$\nu(A) = \int_A f \, d\mu$$

$f = \frac{d\nu}{d\mu}$ (Radon-Nikodym derivative)

Characterizes absolutely continuous measures

$L^p$ Spaces

$L^p(\mu) = \{f \text{ measurable} : \int |f|^p d\mu < \infty\}$ for $1 \leq p < \infty$

$L^{\infty}(\mu) = \{f \text{ measurable} : |f| \leq M \text{ a.e. for some } M\}$

Norm: $\|f\|_p = \left(\int |f|^p d\mu\right)^{1/p}$, $\|f\|_{\infty} = \inf\{M : |f| \leq M \text{ a.e.}\}$

Hölder's Inequality

If $\frac{1}{p} + \frac{1}{q} = 1$ and $f \in L^p$, $g \in L^q$, then: $$\int |fg| \, d\mu \leq \|f\|_p \|g\|_q$$

For $p = 2$: Cauchy-Schwarz inequality

$p, q$ called conjugate exponents

Minkowski's Inequality

For $f, g \in L^p$ and $p \geq 1$: $$\|f + g\|_p \leq \|f\|_p + \|g\|_p$$

Triangle inequality: validates $\|\cdot\|_p$ as a norm

Makes $L^p$ a normed vector space

Completeness of $L^p$

For $1 \leq p \leq \infty$, $(L^p, \|\cdot\|_p)$ is a Banach space

$(L^2, \|\cdot\|_2)$ is a Hilbert space with inner product: $\langle f,g \rangle = \int fg \, d\mu$

Cauchy sequences in $L^p$ converge: $\|f_n - f_m\|_p \to 0 \Rightarrow \exists f \in L^p: f_n \to f$

Convergence Types

Almost everywhere: $f_n \to f$ a.e. if $\mu(\{x: f_n(x) \not\to f(x)\}) = 0$

In measure: $f_n \xrightarrow{\mu} f$ if $\mu(\{|f_n - f| > \epsilon\}) \to 0$ for all $\epsilon > 0$

In $L^p$: $f_n \xrightarrow{L^p} f$ if $\|f_n - f\|_p \to 0$

Hierarchy: $L^p \Rightarrow$ in measure $\Rightarrow$ a.e. (for subsequences)

Product Measures

On $(X \times Y, \mathcal{F} \otimes \mathcal{G})$: $(\mu \times \nu)(A \times B) = \mu(A)\nu(B)$

Product $\sigma$-algebra: $\mathcal{F} \otimes \mathcal{G} = \sigma(\{A \times B : A \in \mathcal{F}, B \in \mathcal{G}\})$

Product of σ-finite measures is unique and σ-finite

Signed Measures

$\nu: \mathcal{F} \to \mathbb{R}$ with $\nu(\emptyset) = 0$ and countable additivity

Hahn decomposition: $X = P \cup N$ (disjoint) with $\nu(A \cap P) \geq 0$, $\nu(A \cap N) \leq 0$

Jordan decomposition: $\nu = \nu^+ - \nu^-$ where $\nu^{\pm}$ are measures

Probability Measure Basics

Probability measure: $(\Omega, \mathcal{F}, \mathbb{P})$ with $\mathbb{P}(\Omega) = 1$

Random variable: measurable function $X: \Omega \to \mathbb{R}$

Expected value: $E[X] = \int X \, d\mathbb{P}$

All measure theory results apply with $\mathbb{P}$ in place of $\mu$

Key Equivalences

Measurability equivalent conditions:

$\{x: f(x) > a\} \in \mathcal{F}$
$\{x: f(x) \geq a\} \in \mathcal{F}$
$\{x: f(x) < a\} \in \mathcal{F}$
$\{x: f(x) \leq a\} \in \mathcal{F}$

Absolute continuity: $\nu \ll \mu$ iff $\mu(A) = 0 \Rightarrow \nu(A) = 0$