Set of bounded elementary (simple) processes is a vector space

Question

I am currently learning about stochastic integration with respect to semimartingales, and I am having trouble understanding every detail of the constructions, since I only learned stochastic integration w.r.t. Brownian motion. I have written the questions I have after I give the definitions.

The setup we use is the following:

Setup: We have a complete probability space $(\Omega, \mathcal{F},P)$ and a filtration $(\mathcal{F}_t)_{t \in [0,T]}$ for some $T \in (0,\infty)$ satisfying the usual condition (i.e. $\mathcal{F}_0$ contains all sets of zero measure and is right-continuous).

In the lecture we came across the following definition of a bounded elementary process:

Definition: A bounded elementary (or simple) process $H$ is a stochastic process of the form $$H= \sum_{i=1}^n h_i I_{((\tau_i,\tau_{i+1}]]}$$ where $h_i \in L^{\infty}(F_{\tau_{I}}; \mathbb{R}^d)$ and $0 \leq \tau_0 \leq \tau_1 \leq \cdots \leq \tau_{n+1} \leq T$ are stopping times. The set of all bounded elementary process is denoted by $bE$.

For two maps $S,T : [0,T] \rightarrow [0,\infty]$ we write the stochastic interval $$((S,T]]=\{(t,\omega) \in [0,T] \times \Omega : S(\omega) < t \leq T(\omega)\}.$$

Now to my questions:

1. how do I see that the space $bE$ is a vector space, if we add the elements point wise? I am really new to the notion of stochastic intervals and not to much familiar with it or how to operate with them.
2. Having established that $bE$ is a vector space, is every elementary process $\mathcal{B}([0,T])\otimes \mathcal{F}$-measurable? And how do I see this?
3. If the answers to question 1 and 2 are yes, I can define the product measure $\lambda_T \times P$ on $([0,T] \times \Omega, \mathcal{B}([0,T])\otimes \mathcal{F})$ and then take the equivalence classes of the processes in $bE$ as a subset of all $\mathbb{R}^d$ valued $([0,T])\otimes \mathcal{F})$-measurable functions (equivalence w.r.t. equality $\lambda_T \times P$ almost everywhere) and obtain a space $bE^0$ of equivalence classes, where an equivalence class always has a representation in the form of a bounded elementary process. Then $bE^0$ is also a vector space and a subset of $L^{\infty}(\lambda_T\times P)$. Then I can define a norm on $bE^0$ by choosing $||\cdot||_{L^{\infty}(\lambda_T \times P)}$ as a norm. Is my argumentation correct?

Thanks a lot in advance!