Let $ M(X, \mathscr{A})$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. We show that for every measurable space $(X,\mathscr{A})$, there exists a $T$-measurable space $(Y,\mathscr{A}^{\prime})$ such that $M_K(X, \mathscr{A})\cong M_K(Y,\mathscr{A}^{\prime})$ and $M_{\infty}(X,\mathscr{A})\cong M_{\infty}(Y,\mathscr{A}^{\prime})$, where $M_K(X,\mathscr{A})$ is the ring of real measurable functions $f\in M(X, \mathscr{A})$ for which $coz(f)$ is a compact element of $\mathscr{A}$, and $M_{\infty}(X,\mathscr{A})$ is the ring of real measurable functions vanishing at infinity on $(X, \mathscr{A})$. Then, we introduce $\sigma$-compact and locally compact measurable spaces. We prove that a $T$-measurable space $(X, \mathscr{A})$ is compact ($\sigma$-compact) if and only if the set $X$ is finite (at most countable) and $\mathscr{A}= \mathcal{P}(X) $. Next, we obtain several equivalent conditions for $ M_{\infty}(X, \mathscr{A})$ to be a regular ring. Finally, we show that if $(X, \mathscr{A})$ is a $T$-measurable space and $ M_{\infty}(X, \mathscr{A})\not=\{0\}$, then there exists a locally compact measurable space $(Y, \mathscr{A}')$ such that $ M_{\infty}(X,\mathscr{A})\cong M_{\infty}(Y,\mathscr{A}^{\prime})$ and $M_K(X,\mathscr{A})\cong M_K(Y,\mathscr{A}^{\prime})$.