Intergal of non negative measurable function on measurable set.

Question

Let $(X,M,\mu)$ is measure space

$f$ is non negative measurable function .

E is measurable set in M .

then $\int_E f=\int \chi_E f$

My Attempt: $\int f=\sup_{0\leq \phi \leq f }{\int \phi }$ where $\phi$ is simple function

$\int_E f=\sup_{0\leq \phi \leq f }{\int \phi }=\sup_{0\leq \chi_E\phi \leq \chi_E f }{\int \phi }$ =$\int \chi_E f$

I have doubt that is above correct? Actually I had proved charactersitics function of type $\chi_E\phi $ but definition required supremum over all possible simple function

Any Help will be appreciated

Answer 1

Note that $\psi$ is a simple function such that $0 \leqslant \psi \leqslant f\chi_E$ on $X$ if and only if $\psi = \phi\chi_E$ where $\phi $ is a simple function such that $0 \leqslant \phi \leqslant f$ on $E$.

Thus,

$$\int_E f = \sup_{0 \leqslant \phi \leqslant f}\int_E\phi = \sup_{0 \leqslant \phi \leqslant f}\int_X\phi\chi_E = \sup_{0 \leqslant \phi\chi_E \leqslant f\chi_E}\int_X\phi\chi_E = \sup_{0 \leqslant \psi \leqslant f\chi_E}\int_X\psi = \int_X f\chi_E$$