Probability Measure

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Let $\Omega$ be a sample space, $\mathscr{F}$ be a Borel field of subsets of $\Omega$.  A probability measure (p.m.) $\mathscr{P}\{\cdot\}$ on $\mathscr{F}$ is a real-valued function with domian $\mathscr{F}$ satisfying 

(1) $\displaystyle\forall\,E\in\mathscr{F}:\,\mathscr{P}\{E\}\geq0$.
(2) If $\{E_j\}$ is a countable collection of (pairwise) disjoint sets in $\mathscr{F}$, then $$\mathscr{P}\left\{\bigcup_J E_j\right\}=\sum_j\mathscr{P}\{E_j\}.$$(3)$\mathscr{P}\{\Omega\}=1$.

These axioms imply the following many properties for all sets in $\mathscr{F}$:

(4) $\mathscr{P}\{E\}\leq1$.
(5) $\mathscr{P}\{\emptyset\}=0$.
(6) $\mathscr{P}\{E^c\}=1-\mathscr{P}\{E\}$.
(7) $\mathscr{P}\{E\cup F\}+\mathscr{P}\{E\cap F\}=\mathscr{P}\{E\}+\mathscr{P}\{F\}$.
(8) $E\subset F\implies\mathscr{P}\{E\}=\mathscr{P}\{F\}-\mathscr{P}\{F\setminus E\}\leq\mathscr{P}\{F\}$.
(9) Monotone property.  $E_n\uparrow E$ or $E_n\downarrow E\implies\mathscr{P}\{E_n\}\rightarrow\mathscr{P}\{E\}$.
(10) Boole's inequality.  $\mathscr{P}\{\bigcup_jE_j\}\leq\sum_j\mathscr{P}\{E_j\}$.

Here is an example to identify a probability measure.  Let $X\geq0$ be a random variable and $\int_\Omega X\,d\mathscr{P}=A$, $0<A<\infty$.  Define for $\Lambda\in\mathscr{F}$, $$\nu(\Lambda)=\frac{1}{A}\int_\Lambda X\,d\mathscr{P}.$$Then
(1) Since $X\geq0$, for all $\Lambda\in\mathscr{F}$, $\nu(\Lambda)\geq0.$
(2) Let $\Lambda_1,\Lambda_2,...$ be disjoint. $$\nu\left(\bigcup_j\Lambda_j\right)=\frac{1}{A}\int_{\bigcup_j\Lambda_j}X\,d\mathscr{P}=\frac{1}{A}\sum_j\int_{\Lambda_j}X\,d\mathscr{P}=\sum_j\nu(\Lambda_j).$$(3) $$\nu(\Omega)=\frac{1}{A}\int_\Omega X\,d\mathscr{P}=\frac{1}{A}A=1.$$Thus, $\nu(\cdot)$ is a probability measure.

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