Conditional Probability Defines Probability Space

From ProofWiki
Jump to navigation Jump to search


Let $\struct {\Omega, \Sigma, \Pr}$ be a measure space.

Let $B \in \Sigma$ such that $\map \Pr B > 0$.

Let $Q: \Sigma \to \R$ be the real-valued function defined as:

$\map Q A = \condprob A B$


$\condprob A B = \dfrac {\map \Pr {A \cap B} }{\map \Pr B}$

is the conditional probability of $A$ given $B$.

Then $\struct {\Omega, \Sigma, Q}$ is a probability space.


It is to be shown that $Q$ is a probability measure on $\left({\Omega, \Sigma}\right)$.

As $\Pr$ is a measure, we have that:

$\forall A \in \Omega: \map Q A \ge 0$

Also, we have that:

\(\ds \map Q \Omega\) \(=\) \(\ds \condprob \Omega B\)
\(\ds \) \(=\) \(\ds \frac {\map \Pr {\Omega \cap B} } {\Pr \left({B}\right)}\)
\(\ds \) \(=\) \(\ds \frac {\map \Pr B} {\map \Pr B}\) Intersection with Universe
\(\ds \) \(=\) \(\ds 1\) as $\map \Pr B > 0$

Now, suppose that $A_1, A_2, \ldots$ are disjoint events in $\Sigma$.


\(\ds \map Q {\bigcup_{i \mathop = 1}^\infty A_i}\) \(=\) \(\ds \frac 1 {\map \Pr B} \map \Pr {\paren {\bigcup_{i \mathop = 1}^\infty A_i} \cap B}\)
\(\ds \) \(=\) \(\ds \frac 1 {\map \Pr B} \map \Pr {\bigcup_{i \mathop = 1}^\infty \paren {A_i \cap B} }\) Intersection Distributes over Union
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^\infty \frac {\map \Pr {A_i \cap B} } {\map \Pr B}\) as $\Pr$ is a measure
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^\infty \map Q {A_i}\)