Question about conditioning on sequences of random variables, where the sequence is related by a function $f$

Question

Let $A_k, B_k$ be two sequences of random variables, where $A_k = f(B_k)$

Then is it true that

$$\mathbb{E}[A_{k+1}|A_0, \ldots, A_k] = \mathbb{E}[f(B_{k+1})|B_0, \ldots, B_k]$$

If so, what is the justification for the equality?

Answer 1

No. In general, the conditioning might be irrelevant (if the sequence is iid), which means that the question is if $E(B)=E(f(B))$ true for any function $f$. In the general case, this is false:

Suppose $B_k=\pm 1$ (w.p. $p\neq 0.5$ to be $1$) and $f(x)=\vert x \vert$. Then $A_k=1$ and $E(A_k)=1$. But $E(B_k)=p-(1-p)\neq 1$.