Let $(Y_n)$ be an adapted sequence of integrable ranodm variables, with the property that for every boudned stopping time $T$ we have $EY_T = 0$. I am asked to show that $EY_n = 0$ for every natural number $n$. I have tried:
Suppose that the stopping time is bounded, say by $K$, then we can write:
$$Y_n = \sum_{i=0}^{k-1} Y_i 1_{\{T = i\}} + 1_{\{T \geq k \}}Y_k$$
by the definition of stopping time, each $1_{\{T = i\}}$ is $\mathcal{F}_{k-1}$ measurable, and similarly $1_{\{T\geq k\}}$ is $\mathcal{F}_{k-1}$ measurable, now I wanted to take $E(Y_n | \mathcal{F}_{k-1})$ and get some sort of conclusion but I can't proceed
Each nonnegative integer $n$ corresponds to a bounded stopping time ($T = n$ a.s.).