I have been struggling with the following problem.
Let $\{X_{n}\}_{n\geq 1}$ be i.i.d. random variables with common distributional measure $\nu$. Let $B\subset\mathbb{R}$ be any Borel measurable set such that $0<\nu(B)<1$, and define
$$\tau=\inf\{n\geq 1\mid\,X_{n}\in B\}$$
If $D\subseteq\mathbb{R}$ is another Borel measurable set, show that
$$\mathbb{P}(X_{\tau}\in D)=\frac{\nu(D\cap B)}{\nu(B)}$$
I have calculated that $\mathbb{P}(\tau=n)=\nu(B)(1-\nu(B))^{n-1}$ and that as a result, $\mathbb{P}(\tau<\infty)=1$. However, I haven't able to make much headway regarding the desired conclusion. I've considered the probabilities $\mathbb{P}(X_{\tau}\in H\mid\tau=n)$, but haven't been able to make much use of them. I don't know a tremendous amount about martingales or stopping times. Leads would be appreciated.
Indeed. Presumably something like this...
$$\begin{align}\mathbb P(\tau=n)&=\Bbb P(n=\inf\{m\geq 1\mid X_m\in B\})\\&=\Bbb P(X_n\in B,\bigcap_{k=1}^{n-1} (X_k\notin B)) \\&=\mathbb P(X_n\in B)\prod_{k=1}^{n-1}\mathbb P(X_k\notin B)\\ &=\nu(B)~(1-\nu(B))^{n-1}\end{align}$$
Now, similarly
$$\begin{align}\Bbb P(X_\tau\in D)&=\sum_{n=1}^\infty\Bbb P(X_n\in D, \tau=n)\\&=\sum_{n=1}^\infty\Bbb P(X_n\in D, n=\inf\{m\geq 1\mid X_m\in B\})\\&=\sum_{n=1}^\infty \Bbb P(X_n\in D, X_n\in B,\bigcap_{k=1}^{n-1}(X_k\notin B))\\&\ddots\end{align}$$