Conditional Properties of Probability
Theorem
Let’s say we have a probability space $( \Omega , \mathcal{F} , P)$ and a sub-sigma field $\mathcal{G} \subset \mathcal{F}$.
- [1] For all $B \in \mathcal{G}$, there is $0 \le P(B | \mathcal{G}) \le 1$.
- [2] Continuity of probability: For a nested sequence $\left\{ B_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{G}$, $$ \lim_{n \to \infty} B_{n} = B \implies P ( B_{n} | \mathcal{G} ) \to P ( B | \mathcal{G} ) \text{ a.s.} $$
- [3] If $\left\{ B_{n} \right\}_{n \in \mathbb{N}}$ is a partition of $\Omega$, $$ P \left( \bigsqcup_{n \in \mathbb{N}} B_{n} | \mathcal{G} \right)= \sum_{n \in \mathbb{N}} P \left( B_{n} | \mathcal{G} \right) $$
- That a sequence of events $\left\{ B_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{G}$ is nested means it has one of the following two properties. $$ \forall n \in \mathbb{N}, B_{n} \subset B_{n+1} \iff B_{n} \subset B_{n+1} \subset \cdots \\ \forall n \in \mathbb{N}, B_{n} \subset B_{n-1} \iff B_{n} \subset B_{n-1} \subset \cdots $$
- A nested sequence may have the following properties for some event $B \in \mathcal{G}$. $$ \forall n \in \mathbb{N}, B_{n} \subset B_{n+1} \land \bigcup_{n \in \mathbb{N}} B_{n} = B \implies \lim_{n \to \infty} B_{n} = B \\ \forall n \in \mathbb{N}, B_{n} \subset B_{n-1} \land \bigcap_{n \in \mathbb{N}} B_{n} = B \implies \lim_{n \to \infty} B_{n} = B $$
- $\bigsqcup$ denotes the symbol for the union of mutually exclusive sets.
Proof
[1]
Since $P$ is a probability, according to the definition of conditional probability and conditional expectation, for all $A \in \mathcal{G}$ $$ \begin{align*} \int_{A} 0 dP \le & \int_{A} P(B | \mathcal{G}) dP \\ =& \int_{A} E ( \mathbb{1}_{B} | \mathcal{G} ) dP \\ =& \int_{A} \mathbb{1}_{B} dP \\ \le & \int_{A} 1 dP \end{align*} $$ $\displaystyle \forall A \in \mathcal{F}, \int_{A} f dm = 0 \iff f = 0 \text{ a.e.}$ hence $0 \le P(B | \mathcal{G}) \le 1$
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[2]
We only need to show that it holds when $\forall n \in \mathbb{N}, B_{n} \subset B_{n+1}$, by setting it as $B_{n} := \Omega \setminus A_{n}$, we can also show it holds when $\forall n \in \mathbb{N}, A_{n} \subset A_{n-1}$. Assuming $\forall n \in \mathbb{N}, B_{n} \subset B_{n+1}$, according to the definition of conditional probability and the Monotone Convergence Theorem for conditional probability, $$ \begin{align*} \lim_{n \to \mathbb{N}} P(B_{n} | \mathcal{G}) \color{red}{=}& \lim_{n \to \infty} E ( \mathbb{1}_{B_{n}} | \mathcal{G} ) \\ \color{blue}{=}& E \left( \lim_{n \to \infty} \mathbb{1}_{B_{n}} | \mathcal{G} \right) \\ =& E \left( \mathbb{1}_{B} | \mathcal{G} \right) \\ \color{red}{=}& P(B | \mathcal{G} ) \end{align*} $$
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[3]
If $\left\{ B_{n} \right\}_{n \in \mathbb{N}}$ is a partition of $\Omega$, then for all $n \in \mathbb{N}$, $\displaystyle \bigsqcup_{k=1}^{n} B_{k} \subset \bigsqcup_{k=1}^{n+1} B_{k}$ hence, according to the continuity of probability in [2], $$ \begin{align*} P \left( \bigsqcup_{n=1}^{\infty} B_{n} | \mathcal{G} \right) =& P \left( \lim_{n \to \infty} \bigsqcup_{k=1}^{n} B_{k} | \mathcal{G} \right) \\ =& \lim_{n \to \infty} P \left( \bigsqcup_{k=1}^{n} B_{k} | \mathcal{G} \right) \\ =& \lim_{n \to \infty} E \left( \mathbb{1}_{\bigsqcup_{k=1}^{n} B_{k}} | \mathcal{G} \right) \\ =& \lim_{n \to \infty} \sum_{k=1}^{n} E \left( \mathbb{1}_{B_{k}} | \mathcal{G} \right) \\ =& \lim_{n \to \infty} \sum_{k=1}^{n} P \left( B_{k} | \mathcal{G} \right) \\ =& \sum_{n=1}^{\infty} P \left( B_{n} | \mathcal{G} \right) \end{align*} $$
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