18.1 Measures

Definition 18.1.1 (Measurable Space).label Let $X$ be a set and $\cm \subset 2^{X}$ be a $\sigma$-algebra, then the pair $(X, \cm)$ is a measurable space.

Definition 18.1.2 (Measure).label Let $(X, \cm)$ be a measurable space and $\mu: \cm \to [0, \infty]$, then $\mu$ is a (countably-additive) measure if

1. (M1)

   $\mu(\emptyset) = 0$.

2. (M2)

   For any $\seq{E_n}\subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}E_n}= \sum_{n \in \natp}\mu(E_{n})$.

In which case, $(X, \cm, \mu)$ is a measure space.

If $\mu: \cm \to [0, \infty]$ instead satisfies (M1) and

1. (M2’)

   For any $\seqf{E_j}\subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{j = 1}^n E_j}= \sum_{j = 1}^{n} \mu(E_{j})$.

then $\mu$ is a finitely-additive measure.

Definition 18.1.3 (Null Set).label Let $(X, \cm, \mu)$ be a measure space, then $E \in \cm$ is $\mu$-null/null if $\mu(E) = 0$.

Definition 18.1.4 (Almost Everywhere).label Let $(X, \cm, \mu)$ be a measure space. A statement holds $\mu$-almost everywhere/$\mu$-a.e./a.e. if it is false on a subset of a $\mu$-null set.

Proposition 18.1.5 ([Theorem 1.8, Fol99]).label Let $(X, \cm, \mu)$ be a measure space, then:

1. (1)

   For any $E, F \in \cm$ with $E \subset F$, $\mu(E) \le \mu(F)$.

2. (2)

   For any $\seq{E_n}\subset \cm$, $\mu\paren{\bigcup_{n \in \natp}E_n}\le \sum_{n \in \natp}\mu(E_{n})$.

3. (3)

   For any $\seq{E_n}\subset \cm$ with $E_{n} \subset E_{n+1}$ for all $n \in \nat$, $\mu\paren{\bigcup_{n \in \nat}E_n}= \limv{n}\mu(E_{n})$.

4. (4)

   For any $\seq{E_n}\subset \cm$ with $E_{n} \supset E_{n+1}$ for all $n \in \nat$ and $\mu(E_{1}) < \infty$, $\mu(\bigcap_{n \in \natp}E_{n}) = \limv{n}\mu(E_{n})$.

5. (5)

   For any $\seq{E_n}\subset \cm$,

   \[\mu\paren{\liminf_{n \to \infty}E_n}\le \liminf_{n \to \infty}\mu(E_{n})\]

   
   
6. (6)

   For any $\seq{E_n}\subset \cm$ with $\mu\paren{\bigcup_{n \in \nat}E_n}< \infty$,

   \[\mu\paren{\limsup_{n \to \infty}E_n}\ge \limsup_{n \to \infty}\mu(E_{n})\]

   
   

Theorem 18.1.6 (Dynkin’s Uniqueness Theorem).label Let $(X, \cm)$ be a measurable space, $\mathcal{P}\subset \cm$ be a $\pi$-syste, and $\mu, \nu: \cm \to [0, \infty]$ be measures. If

1. (a)

   $\sigma(\mathcal{P}) = \cm$.

2. (b)

   $\mu(E) = \nu(E)$ for all $E \in \mathcal{P}$.

3. (c)

   There exists $\seq{E_n}\subset \mathcal{P}$ such that $E_{n} \upto X$ and $\mu(E_{n}) < \infty$ for all $n \in \natp$.

then $\mu = \nu$.

Lemma 18.1.7 (First Borel-Cantelli Lemma).label Let $(X, \cm, \mu)$ be a measure space and $\seq{E_n}\subset \cm$. If $\sum_{n \in \natp}\mu(E_{n}) < \infty$, then

Definition 18.1.8 (Pushforward Measure).label Let $(X, \cm, \mu)$ be a measure space, $(Y, \cn)$ be a measurable space, and $f: X \to Y$ be a $(\cm, \cn)$-measurable map, then:

1. (1)

   The mapping

   \[f_{*}\mu: \cn \to [0, \infty] \quad A \mapsto \mu\bracs{f \in A}\]

   
   

   is a measure on $(Y, \cn)$.

2. (2)

   For any Banach space $E$ and $g \in L^{1}(f_{*}\mu; E)$,

   \[\int g df_{*}\mu = \int g \circ f d\mu\]