The integral with regard do $\mathrm{d}M^*$ is a pathswise Stieltjès integral, so the question is an analysis problem.

Let $f : \mathbb{R}_+ \to \mathbb{R}$ be any continuous function, $F$ its current maximum, and $\mu$ the Stieltjès measure associated to $F$

One checks that $F$ is also continuous, so $O := \{s \in \mathbb{R}_+ : F(s)-f(s)>0\}$ is open subset in $\mathbb{R_+}$ and contained in $\mathbb{R_+}^*$ since $F(0)=f(0)$. Hence it is an at most countable union of disjoint open intervals. On each one of these open intervals, $F$ remains constant, so the $\mu$-measure of this interval is $0$. As a result $\mu(O)=0$, so $F-f$ is null $\mu$-almost everywhere and the integral $\int(F-f) \mathrm{d}\mu$ is $0$.

The integral with regard do $\mathrm{d}M^*$ is a pathwise Stieltjès integral, so the question is an analysis problem.

Let $f : \mathbb{R}_+ \to \mathbb{R}$ be any continuous function, $F$ its current maximum, and $\mu$ the Stieltjès measure associated to $F$

One checks that $F$ is also continuous, so $O := \{s \in \mathbb{R}_+ : F(s)-f(s)>0\}$ is open subset in $\mathbb{R_+}$ and contained in $\mathbb{R_+}^*$ since $F(0)=f(0)$. Hence it is an at most countable union of disjoint open intervals. On each one of these open intervals, $F$ remains constant, so the $\mu$-measure of this interval is $0$. As a result $\mu(O)=0$, so $F-f$ is null $\mu$-almost everywhere and the integral $\int(F-f) \mathrm{d}\mu$ is $0$.