Rings of real measurable functions vanishing at infinity on a measurable space

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=\{۰\}, 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}).

Keywords:

Measurable space , compact , \sigma-compact , sigma-frame , Regular ring