Random variable measurable with respect to stopped sigma algebra

Let $$(\Omega,\mathscr A,\mathscr F_n,\mathbb P)$$ be a filtered probability space and $$\tau$$ be an $$\mathscr F_n$$ stopping time. $$\mathscr F_n$$ is discretely indexed.

Show that $$X:\Omega \to \mathbb R$$ is measurable with respect to the stopped sigma algebra $$\mathscr F_\tau$$ if and only if $$X(\omega)=\sum_{n=1}^\infty Y_n(\omega)1_{\{\tau =n\}}(\omega).$$ for an $$\mathscr F_n$$ adapted process $$Y_n$$.

Showing $$X$$ of that form is $$\mathscr F_\tau$$ measurable is more or less straightforward. $$\tau$$ is $$\mathscr F_\tau$$ measurable and indicators on measurable sets are measurable. $$Y_n$$ is adapted and taking intersection gives the result. For the other direction, how can I find the $$Y_n$$?

We need to assume $$\tau<\infty$$ almost surely. (If $$\mathbb P(\tau=\infty)>0$$, then $$\tau$$ itself is an $$\mathscr F_\tau$$-measurable random variable which cannot be expressed in the given form.)
For each $$n\in\mathbb N$$, let $$Y_n = X1_{\{\tau=n\}}$$. Since $$X$$ is $$\mathscr F_\tau$$-measurable, $$Y_n$$ is $$\mathscr F_n$$-measurable, that is, $$\{Y_n\}$$ is adapted. Clearly, we have
$$X = X\sum_{n=1}^\infty 1_{\{\tau=n\}} =\sum_{n=1}^\infty X1_{\{\tau=n\}} = \sum_{n=1}^\infty Y_n=\sum_{n=1}^\infty Y_n1_{\{\tau = n\}},$$
• I suppose technically yes, but often when we write an expression for $X$ there is an implicit "almost surely" involved, so it makes no difference in assuming only $\tau<\infty$ almost surely. Apr 20, 2020 at 14:25