Gambling interpretation of conditional probability In Billingsley, when defining conditional probability the following property has been given a gambling interpretation : 
$$ \int_G P[A||\mathscr{G}]dP = P(A \cap G), G \in  \mathscr{G} $$
where at the end it says that the above property implies that the observer's strategy is fair. Here the entry fee to the game is $P(A||\mathscr{G})$. I could not understand the meaning/use of such interpretation.  
 A: In Billingsley's example the gambler is deciding whether or not to bet after the outcome of $\mathcal G$ is learned, where the payoff depends on $A$ being in $\mathcal G$.  The gambler (I'll call him Kenny) looks at whether or not $G$ happened (this evidently be determined after $\mathcal G$ occurs), then decides whether or not to place a bet that $A$ happened.*  Fairness is determined by computing expected value.  Kenny gains $1-P[A||\mathcal G]$ if $A$ occurs and loses $P[A||\mathcal G]$ if $A$ doesn't occur: $$\bigl(1-P[A||\mathcal G]\bigr)I_A - P[A||\mathcal G] I_{A^c} = I_A - P[A||\mathcal G](I_A + I_{A^c}) = I_A - P[A||\mathcal G].$$  If $P[A||\mathcal G]$ were a fair entry fee then Kenny's gain for each $G\in \mathcal G$ should be $$0 = \int_G \bigl( I_A - P[A||\mathcal G]\bigr) \,dP= P(A\cap G) - \int_G P[A||\mathcal G]\,dP$$ and the result follows. 
{*} Tantamount to betting on whether or not it's raining outside your office building by seeing whether or not your coworkers are wet as they come back from lunch.
