A function $f: E \rightarrow \mathbb{X}$ is called simple if it can be represented as
$$f(\omega)=\sum_{i=1}^k I_{E_i}(\omega) g_i$$
for some finite $k, E_i \in \mathscr{B}$ and $g_i \in \mathbb{X}$.
Any simple function $f(\omega)=\sum_{i=1}^k I_{E_i}(\omega) g_i$ with $\mu\left(E_i\right)<\infty$ for all $i$ is said to be integrable and its Bochner integral is defined as
$$\int_E f d \mu=\sum_{i=1}^k \mu\left(E_i\right) g_i$$
Question: now I want to show that this definition does not depend on the particular representation of $f$.
My attempt: To show that the definition of the Bochner integral does not depend on the particular representation of the simple function $f$, we need to prove that different representations of the same simple function will yield the same integral. Let's consider two different representations of the same simple function $f$:
$f(\omega)=\sum_{i=1}^k I_{E_i}(\omega) g_i=\sum_{j=1}^m I_{F_j}(\omega) h_j$
where $\{E_i\}_{i=1}^k$ and $\{F_j\}_{j=1}^m$ are partitions of $E$, and $g_i, h_j \in \mathbb{X}$.
How to prove the above equation?