Let $E$ be a Banach space. I want to discuss the relationship of $\mathcal{S}(\mathbb{R},E)$, which, to me, is the set of functions that are regulated on any compact interval, and $\mathcal{L}_0(\mathbb{R},\beta_1,E)$, which, to me, is the set of $\beta_1$-measurable functions from $\mathbb{R}$ to $E$. ($\beta_1=\lambda_1|_{\mathcal{B}(\mathbb{R})}$)
I think, $\mathcal{S}(\mathbb{R},E)\subseteq\mathcal{L}_0(\mathbb{R},\beta_1,E)$: For $f\in\mathcal{S}(\mathbb{R},E)$ and $n\in\mathbb{N}$ we can find a sequence of step functions $(f^n_j)_{j\in\mathbb{N}}$ such that $f_j\to f$ on $[-n,n]$. We can extend $f^n_j$ to be a $\beta_1$-simple function on $\mathbb{R}$, $j\in\mathbb{N}$, such that $f_j^n\to 1\cdot\chi_{[-n,n]}\cdot f$. So, $f_n:=1\cdot\chi_{[-n,n]}\cdot f\in\mathcal{L}_0(\mathbb{R},\beta_1,E)$, and $f=\lim_{n\to\infty}f_n\in\mathcal{L}_0(\mathbb{R},\beta_1,E)$. Is my reasoning alright?
I think, $\mathcal{S}(\mathbb{R},E)\not\supseteq\mathcal{L}_0(\mathbb{R},\beta_1,E)$: Consider $\chi_\mathbb{Q}$. Is my reasoning alright?