# Intergal of non negative measurable function on measurable set.

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

• What is $\int_{XE}f$? – herb steinberg May 31 '19 at 17:00
• What is your definition of $\int_E f$? The usual definition is $\int_E f=\int\chi_E f$. – David C. Ullrich May 31 '19 at 17:16
• Dear Sir , I had mentioned definition in my attempt: – MathLover May 31 '19 at 17:17
• $\int_E f=\sup_{0\leq \phi \leq f }{\int_E \phi }$ – MathLover May 31 '19 at 17:18

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