I have a problem proving the following theorem...

Let $\displaystyle T_1, T_2, ...$ be independent random variables of exponential distributions with parameters $\displaystyle q_i > 0$. Let $\displaystyle T(\omega) = \inf_{j \geq 1} T_j$. Provided $\displaystyle \sum_{i=1}^\infty q_i =q < \infty$ prove that for every $\displaystyle \omega \in \Omega$ (except for zero-probability set) exists exactly one index $\displaystyle j = J(\omega)$ such that $\displaystyle T(\omega) = T_j(\omega)$. What is more $\displaystyle T$ and $\displaystyle J$ are independent, and $\displaystyle P(J = j) = \frac{q_j}{q}$.