This is part of the proof of Theorem 2 from the following post : https://almostsuremath.com/2010/05/17/sdes-under-changes-of-time-and-measure/
Let $M$ be a continuous semimartingale. Consider the following identity $$F(M_t) = F(M_0)+\int_0^t F'(M)dM + \frac{1}{2} \int_0^t f(M)d[M]$$ where $F(x)=\int_0^x (x-y)f(y)dy$.
Then this identity holds for any continuous function $f:\mathbb{R}\to \mathbb{R} $ as $F$ is twice continuously differentiable, $F'(x)= \int_0^x f(y)dy$ and $F''(x)=f(x)$, this is just Ito's lemma.
Then we can apply the monotone class theorem and use the fact that continuous functions approximate bounded measurable functions and dominated convergence theorems for Lebesgue-Stieltjes integral and stochastic integrals to extend that the identity to all bounded measurable $f$.
My question is : How do we extend this identity to all nonnegative and locally integrable $f$ by monotone convergence?
Firstly, I am not aware of any approximation results for locally integrable functions via bounded measurable functions.
And secondly, to use monotone convergence, as $F$ itself is defined in terms of Lebesgue integral, and we have a stochastic integral and Lebesgue-Stieltjes integrals on the right hand side, we would need to use dominated or bounded convergence theorems on the right hand side and on the left hand side as well to derive the identity.
However, I do not know why $f$ being locally integrable would give the integrability of $\int_0^t F'(M)dM$ and $\int_0^t f(M) d[M]$.
I would greatly appreciate if anyone could explain how this convergence works out here. The argument is stated at the bottom of the following:
