Does this lemma have a name?

Question

Does the following lemma have a name? I'm asking this because the following lemma is very useful and easy to prove. So it's very likely that it has a name in the literature.

Let $(X,\Sigma )$ be a measure space and $\mathcal{M(\Sigma) }$ be the set that contains exactly all measurable functions $f:X\to \mathbb{R}$. Suppose that $\mathcal{M}\subseteq \mathcal{M}(\Sigma)$ is such that

1. $\chi _E\in\mathcal{M}$ (characteristic function of $E$) for all $E\in \Sigma $
2. $af+bg\in\mathcal{M}$ for all $f,g\in\mathcal{M}$ and $a,b\in \mathbb{R}$
3. If $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{M}$ that converges pointwise, then $\lim_{n\to\infty }f_n\in \mathcal{M}$.

Then $\mathcal{M}=\mathcal{M}(\Sigma )$.

To prove the above proposition you need to use that every measure function is a pointwise limit of simple functions.

Thank you for your attention!