I have been self-studying the book on Stochastic Differential Equations and Diffusion Processes by Watanabe and Ikeda. Here is something I dont really follow
An increasing integrable process $A_t$(A is adapted to $\mathcal{F}_t$, integrable for all $t$ with right continuous increasing paths) is natural if $E\int_0^t m_s dA_s = E\int_0^t m_{s-} dA_s$ for every bounded martingale(we always use the right continuous modification of this bounded martingale which always exists as a consequence of the Doob's Upcrossing inequality)
Claim:If an integrable increasing process $A$ is continuous then its natural
Proof(In book)):Since $M$ is right continuous the integral $\int_0^t \mathbb{1}_{[M_s \neq M_{s-}]} dA_s$=0.a.s This implies that $\int_0^t M_sdA_s=\int_0^t M_{s-}dA_s$
My Attempt So $\int_0^t \mathbb{1}_{[M_s \neq M_{s-}]} dA_s$=0 is true because if we think of this integral as a lebesgue stieltjes integral then since for every $\omega$, $t \mapsto A_t(\omega)$ induces a unique measure and since the set $\{s\in(0,t):M_s(\omega) \neq M_{s-}(\omega)\}$ has lebesgue measure $0$, we have that $\int_0^t \mathbb{1}_{[M_s(\omega) \neq M_{s-}(\omega)]} dA_s(\omega)=0$.(It has lebegue measure zero because a right continuous function has atmost countably many discontinuities).
Is this reasoning correct or am I wrong?
I would be grateful if you could help me out here