Definition (Measure space)

Let $(X,\mathcal{A})$ be a measurable space and define a measure $\mu$ on $\mathcal{A}$, the triple $(X,\mathcal{A},\mu)$ is called a measure space if

1. Monotonicity: $\mu(A)\le\mu(B)$ if $A\subseteq B$
2. If $\mu(A)<\infty$ then $\mu(B\backslash A)=\mu(B)-\mu(A)$
3. For every countable family of subsets $(A_{j})_{j\in\mathbb{N}}$ from $\mathcal{A}$ for which $A_{j}\subset A_{j+1}$ (i.e. $A_{j}$ is increasing in $j$) we have that $$\mu\left(\bigcup_{j\in\mathbb{N}}A_{j}\right)=\lim_{j\to\infty}\mu(A_{j})$$
4. For every countable family of subsets $(A_{j})_{j\in\mathbb{N}}$ from $\mathcal{A}$ for which $A_{j}\supset A_{j+1}$ (i.e. $A_{j}$ is decreasing in $j$) we have that $$\mu\left(\bigcap_{j\in\mathbb{N}}A_{j}\right)=\lim_{j\to\infty}\mu(A_{j})$$