Conditional Probability Defines Probability Space

Theorem
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:


 * $\map Q A = \condprob A B$

where:


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

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

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

Proof
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:

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

Then: