Events of a Probability Space: Subsets of Ω, Real Probabilities, Complements and Tail Unions

In the setting of Probability Spaces: Notation, let $(\Omega,\mathcal{F},P)$ be a probability space, let $E\in\mathcal{F}$ be an event, and let $(A_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{F}$. Elements of the type $\mathrm{Real}$ of The Real Numbers as a Type (i) carry the $0$, $1$, $-$ and $\le$ of The Real Numbers as a Type (ii), the literals $0$ and $1$ read as in The Real Numbers as a Type (iii). For $n\in\mathbb{N}$, let $T_{n}$ be the class of all $\omega$ such that $\omega\in A_{k}$ for some $k\in\mathbb{N}$ with $n\le k$, the order of $\mathbb{N}$; and let $L$ be the class of all $\omega\in\Omega$ such that for every $n\in\mathbb{N}$ there is $k\in\mathbb{N}$ with $n\le k$ and $\omega\in A_{k}$.

(i) Events are subsets of $\Omega$. $E$ is a subclass of $\Omega$: $E\subseteq\Omega$.

(ii) Probabilities are real. $P(E)\in\mathbb{R}$, and the element $p$ of $\mathrm{Real}$ with set $P(E)$ satisfies

$$0\le p\le1 .$$

(iii) Complements. The difference $\Omega\setminus E$ lies in $\mathcal{F}$, and with $p$ as in (ii) and $q$ the element of $\mathrm{Real}$ with set $P(\Omega\setminus E)$,

$$q=1-p .$$

(iv) Tail unions. Let $m,n\in\mathbb{N}$. A set $\omega$ lies in $T_{n}$ if and only if $\omega\in A_{k}$ for some $k\in\mathbb{N}$ with $n\le k$; $T_{n}\in\mathcal{F}$; and if $m\le n$, then $T_{n}\subseteq T_{m}$. Moreover there is a sequence in $\mathcal{F}$ whose $n$-th term is $T_{n}$ for every $n\in\mathbb{N}$.

(v) Infinitely many events. A set $\omega$ lies in $L$ if and only if $\omega\in\Omega$ and for every $n\in\mathbb{N}$ there is $k\in\mathbb{N}$ with $n\le k$ and $\omega\in A_{k}$; and $L\in\mathcal{F}$.