Let $f: \mathbb{R} \to \mathbb{R}$ a function with $f \in L^2_{[0,1]}$ for each $t \ge 0$, $(B_t)_t$ a Brownian motion process on a probability space $\Omega$ and $(\mathcal{F}_t)_t$ the canonical filtration with respect to $(B_t)_t$.
I want to show that $M_t := \int_0 ^t f(s) dB_s$ is a martingale.
The main two problems are how to see that :
-$(M_t)_t$ is adapted with pespect to filtration $(\mathcal{F}_t)_t$
-and the most crucial point, the fact that for $r \le t$ we have the martingale property for conditional expectation, i.e. :
$$\mathbb{E}(M_t \vert \mathcal{F}_r) =\mathbb{E}(\int_0 ^t f(s) dB_s \vert \mathcal{F}_r) = \int_0 ^r f(s) dB_s = M_r$$