Let $f:[0,1] \to \mathbb R$ be Lipschitz with Lipschitz constant $K$. Let $f_n$ be the function obtained from $f$ through linear interpolation through the points $({k2^{-n}})_{0\le k\le2^{n}}$ Define
$M_n(x)=\begin{cases} f_n'(x), & \text{if $x$ $\neq$ $k2^{-n}, k=0...2^{n}$} \\ \lim_{x_m \to x} f_n'(x_m), & \text{$x=k2^{-n}, k=0...2^{n}$} \\ f(1) & \text{$x=1$}\\ \end{cases}$
I was able to show that $M_n$ is a martingale and now I want to show that
Show theres exists a bounded measurable function $g:[0,1] \to \mathbb R$ such that for all $x$
$f(x)-f(0)=\int_0^x g(s)ds$
My attempt: One can show that $M_n$ is $L^1$bounded, so it is uniformly integrable and we therefore know that $M_n$ converges almost surely and in $L^1$ to a r.v lets call it $M_\infty$. I now want to say something like
$$\int_0^x M_n(x)=f_n(x)-f_n(0).$$
Taking $n \to \infty$ and using $L^1$ convergence
$$\int_0^x M_\infty(x)dx=f(x)-f(0).$$
But it seems a little vague. Can anybody help me with this part?