Measure

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Definition (Measure)

Let $(X,\mathcal{A})$ be a measurable space, a measure on $\mathcal{A}$ is a map $\mu:\mathcal{A}\to\overline{\mathbb{R}}^{+}$ such that

$\exists E\in\mathscr{M}$ such that $\mu(E)<\infty$
Countable Additivity: $\mu\left(\bigsqcup_{j=1}^{\infty}A_{j}\right)=\sum\limits_{j=1}^{\infty}\mu(A_{j})$ for every countable family $(A_{j})_{j\in\mathbb{N}}$ of pairwise disjoint sets from $\mathcal{A}$ .

Theorem (1.19)

Let $(X,\mathcal{M},\mu)$ be a Measure Space, then

$\mu(\emptyset)=0$
Finite Additivity: $\mu\left(\bigsqcup_{i=1}^{n}A_{i}\right)=\sum_{i=1}^{n}\mu(A_{i})\iff A_{i}\cap A_{j}=\emptyset,\ \forall i\not=j\text{, i.e. pairwise disjoint}$
Monotonicity: For $A,B\in\mathcal{M}$ $A\subseteq B\implies \mu(A)\le\mu(B)$
If $(A_{n})_{n\in\mathbb{N}}\subset \mathcal{M}$ , $A_{1}\subseteq A_{2}\subseteq\dots$ then $\mu\left( \bigcup_{n=1}^{\infty}A_{n} \right)=\lim_{ n \to \infty } \mu(A_{n})$
If $(A_{n})_{n\in\mathbb{N}}\subset \mathcal{M}$ , $A_{1}\supseteq A_{2}\supseteq\dots$ and $\exists k\ge 1$ s.t. $\mu(A_{k})<\infty$ then $\mu\left( \bigcap_{n=1}^{\infty}A_{n} \right)=\lim_{ n \to \infty } \mu(A_{n})$