In real analysis, we know
Consider any mapping $f: X \rightarrow Y$. If $(X, \mathcal{N})$ and $(Y, \mathcal{N})$ are measurable spaces, a mapping $f: X \rightarrow Y$ is called $(\mathcal{M}, \mathcal{N})$-measurable, or just measurable when $\mathcal{M}$ and $\mathcal{N}$ are understood, if $f^{-1}(E) \in \mathcal{N}$ for all $E \in \mathcal{N}$.
In other book, I'v seen another definition of measurable -- "strongly measurable":
Suppose that $X$ is a real Banach space with norm $\|\cdot\|$. A simple function $f:(0, T) \rightarrow X$ is a function of the form $$ f=\sum_{j=1}^{N} c_{j} \chi_{E_{j}} $$ where $E_{1}, \ldots, E_{N}$ are Lebesgue measurable subsets of $(0, T)$ and $c_{1}, \ldots, c_{N} \in X$.
Definition : A function $f:(0, T) \rightarrow X$ is strongly measurable, or measurable for short, if there is a sequence $\left\{f_{n}: n \in \mathbb{N}\right\}$ of simple functions such that $f_{n}(t) \rightarrow f(t)$ strongly in $X$ (i.e. in norm) for $t$ a.e. in $(0, T)$.
Q: Dose these two definitions coincide? I also want to know the origin of the "strongly measurable", because I think the definition in real analysis is adequate to describe all situation.