Question about integrable functions and integrands

Question

Suppose that $f \in L(X, \mathbb{X}, \mu)$ and that

$\lambda(E)=\int_E f d\mu$ for all $E \in \mathbb{X}$.

Show that $\lambda(E) \geq 0$ for all $ E \in \mathbb{X}$ if and only if $f(x) \geq 0$ for almost all $x \in \mathbb{X}$. Moreover, $\lambda(E)=0$ for all $E$ if and only if $f(x)=0$ for almost $x \in \mathbb{X}$.

I assume that one can argue if $f(x) \geq 0 $ then $\int_E f d\mu \geq 0$. However I am unsure how to do the opposite implication. Further I assume that the second part is the same as the proof of $\int f d\mu=0$ iff $f(x)=0$ $\mu$ a.e. which can be found in most textbooks on measure theory.

Answer 1

Assume $\lambda(E) \geq 0$ and let $f$ be a corresponding function. Assume $f$ is negative on a set $N$ and proceed to prove that $N$ must always be of zero $\mu$-measure.

Start by proving $\int_N f \, d\mu= 0$ which is straigthforward and conclude $\mu(N) = 0$ using the same arguments that were discussed in the solutions of your previous post

math.stackexchange.com/questions/630029/question-of-lebesgue-integrals-to-show-functions-are-the-same-mu-a-e

Further I assume that the second part is the same as the proof of $\int f d\mu=0$ iff $f(x)=0$ μ a.e. which can be found in most textbooks on measure theory.

Yup, seems about right, even if you should be more clear about the fact that we require the integral to vanish on all measurable sets and not just $X$.