Why are simple integrands adapted?

Question

Let be a stochastic process $H$ given by $H(t)=\sum_{k=1}^n H_k\mathbb 1_{(t_{k-1},t_k]}, t\in [0,T], n\in \mathbb N$ and suitable constants $0=t_0<\ldots <t_n=T$ and bounded $F(t_{k-1})$ measurable random variables $H_k, k=1,..,n$, i.e. a simple integrand.

Why is this process adapted wrt to the filtartion $F$, by definition it is just $F(t_{k-1}) $ measuarble. Could someone explain this ?

Answer 1

Let $t$ be an element of $(0,T]$. There exists a $k_0\in \{0,\dots,n-1\}$ such that $t_{k_0}\lt t\leqslant t_{k_0+1}$. Then $ H(t)=H_{k_0+1} $ and $H_{k_0+1}$ is $\mathcal F_{t_{k_0+1-1}}$-measurable, that is, $\mathcal F_{t_{k_0}}$-measurable and since $t_{k_0}\lt t$, $\mathcal F_{t_{k_0}}\subset \mathcal F_t$ hence $H(t)$ is $\mathcal F_t$-measurable.