Suppose $M(t), 0 \leq t \leq T$, is a martingale with respect to some filtration $\mathcal{F}(t), 0 \leq t \leq T$. Let $\Delta(t), 0 \leq t \leq T$, be a simple process adapted to $\mathcal{F}(t)$ (i.e., there is a partition $\Pi=\left\{t_0, t_1, \ldots, t_n\right\}$ of $[0, T]$ such that, for every $j, \Delta\left(t_j\right)$ is $\mathcal{F}\left(t_j\right)$-measurable and $\Delta(t)$ is constant in $t$ on each subinterval $\left[t_j, t_{j+1}\right)$ ). For $t \in\left[t_k, t_{k+1}\right)$, define the stochastic integral
$$
I(t)=\sum_{j=0}^{k-1} \Delta\left(t_j\right)\left[M\left(t_{j+1}\right)-M\left(t_j\right)\right]+\Delta\left(t_k\right)\left[M(t)-M\left(t_k\right)\right] .
$$
We think of $M(t)$ as the price of an asset at time $t$ and $\Delta\left(t_j\right)$ as the number of shares of the asset held by an investor between times $t_j$ and $t_{j+1}$. Then $I(t)$ is the capital gains that accrue to the investor between times 0 and $t$. Show that $I(t), 0 \leq t \leq T$, is a martingale.