Let $B_t$ be a martingale and two times $k\leq t$. I'm looking for a condition on Itô's lemma such that $f(t,B_t)$ is a martingale. Now, there is a part on the solution that I'd like that
$$ \mathbb{E}\left[\int_0^t \frac{d}{dB_t} f(s,B_s)dB_s \middle|\ \mathcal{F}_k\right]= \int_0^k f(s,B_s)dB_s $$
and in general, taking $g$ a specific function,
$$ \mathbb{E}\left[\int_0^t g(s,B_s)dB_s \middle|\ \mathcal{F}_k\right]= \int_0^k g(s,B_s)dB_s $$
Is it true for any $g$? why? I've been trying by definition of the lebesgue integral, but I'm not sure how to proceed.