First of all, sry for the title. I just couldn't figure out any better description for this weird problem:
Let $X$ be a bounded real r.v. and $(A_t)_{t\geq0}$ an increasing bounded process (hence $A_{\infty}\leq M$ a.s for some real $M$.). EDIT: And assume additionally that $A(0)=0$ a.s.
Show that $\mathbb{E}[XA_{\infty}]=\mathbb{E}[\int_{0}^{\infty}\mathbb{E}[X|\mathcal{F}_t]\mathbb{d}A_t]$
Since this exercise is really different from others I have done and differs from the stuff I already worked with I have to admit that I got no clue what to do. Especially the integral with respect to $A_t$ bothers me...
Nevertheless I tried to figure out a simple case. $X=5$ and $A_t=t-1$, $t\in[0,1]$, $A_t=0$ if $t\geq1$. Here the LHS becomes $0$ and the RHS (not sure if that's true) $5\int_0^1\mathbb{d}A_t=5\neq0$??
I'm really confused...maybe we additionally need positivity of $A$?
Suppose that $A_t$ is $\mathcal{F}_t$-measurable. Otherwise the statement is in general not correct; for example if $X$ is independent of $(\mathcal{F}_t)_{t \geq 0}$, but not independent of $(A_t)_{t \geq 0}$.
Hints Since $X$ as well as the process $A$ are bounded it suffices to show
$$\mathbb{E}(X \cdot A_T) = \mathbb{E} \left( \int_0^T \mathbb{E}(X \mid \mathcal{F}_t) \, dA_t \right)$$
for any $T \geq 0$. As $A$ is increasing, we have
$$X \cdot A_T = \sum_{j=1}^n X \cdot (A_{t_j}-A_{t_{j-1}})$$
for any partition $0=t_0<\ldots<t_n = T$. Conclude that
$$\mathbb{E}(X \cdot A_T) = \mathbb{E} \left( \sum_{j=1}^n \mathbb{E}(X \mid \mathcal{F}_{t_{j}}) \cdot (A_{t_j}-A_{t_{j-1}}) \right)$$
Finally, use the definition of the Riemann-Stieltjes integral.