could any one help me to solve the problem?
$E\subseteq M\subseteq \mathbb {R}$, $M$ is measurable given that outer measure $m$ of $M<\infty$, we need to show $E$ is measurable iff $m(M)=m(E)+m(M\setminus E)$
could any one help me to solve the problem?
$E\subseteq M\subseteq \mathbb {R}$, $M$ is measurable given that outer measure $m$ of $M<\infty$, we need to show $E$ is measurable iff $m(M)=m(E)+m(M\setminus E)$
Notice that $\Rightarrow$ is trivial, so you only really need to show the other one.
Because $M$ is measurable, we can assume that $A\subseteq M$, and because it is of finite measure, we can assume that so are all the involved sets.
We want to have $\inf\{ m(U)\vert U\supseteq A\}=\inf\{ m(U)\vert U\supseteq A\cap E\}+\inf\{ m(V)\vert V\supseteq A\setminus E \}$ which can be rewritten as $\inf\{ m(U)\vert U\supseteq A\}=\inf\{ m(U)+m(V)\vert U\supseteq A\cap E,V\supseteq A\setminus E \}$ and further $\inf\{ m(U)\vert U\supseteq A\}=\inf\{ m(U\cup V)+m(U\cap V)\vert U\supseteq A\cap E,V\supseteq A\setminus E \}$ But notice that by the assumption for each $n$, we have $U_n,V_n$ measurable such that $U_n\supseteq E$ and $V_n\supseteq M\setminus E$ and $m(U_n\cap V_n)<1/n$ (to show this, you need to use finiteness).
In particular, can assume that $(U\cap V)$ is of very small measure in the right hand side, yielding the equality. I will leave the technical details to you.