On Building Random Variables of a Given Distribution

Given $(X_t)_{t\geqq1}$, independent random variables on some measurable space $(I, \mathscr{B})$ with the same distribution $m$, and a positive function $f$ of $L^1(m)$ with $\|f\|_1 = 1$, this paper studies how to build a stopping time $T$ with respect to the $\sigma$-fields $\mathscr{F}_t$ generated $X_1, X_2, \cdots, X_t$, such that the distribution of $X_T$ in $(I, \mathscr{B})$ is exactly $f dm$.