# probability of an event with countably many sample points

The following is Problem 1.9 from A Modern Approach to Probability Theory.

Let $A$ be an event which contains countably many sample points. Assume that each of the sample points in $A$ is an event. Find a formula for the probability of $A$ in terms of the probabilities of the sample points in $A$.

I was thinking if $(\Omega,\mathcal{F},\mathbb{P})$ is our probability space and $A=\{x_1,x_2,\ldots\}$ where $\{x_i\}\in\mathcal{F}, \forall i\in\mathbb{N}$

I was thinking of something simple like $\mathbb{P}(A)=\sum_{i=1}^\infty \mathbb{P}(\{x_i\})$ but I am not sure since this seems too simplistic.

• Looks okay to me. I would choose for $P(A)=\sum_{a\in A} P(\{a\})$ to include'finite' and 'empty'. – drhab Sep 1 '14 at 14:34
• You could make it notationally more complicated if you like :-): $E[1_A] = E[\sum_i 1_{\{x_i\} }] = \sum_i E[1_{\{x_i\}}]$. – copper.hat Sep 1 '14 at 15:08