I was trying to solve an exercise in probability theory and I wondered if the following is true:
Let $(\Omega,\mathfrak{A},\mu)$ be a measure space, $A\in\mathfrak{A}$ and $f\in L^1(\mu)$ nonnegative. Then we have that $\int_\mu{\chi_A f \text{d}\omega}\leq \mu(A)||f||_{L^1(\mu)}$.
I tried to find a counterexample on the real line but failed to find one. Maybe there is a really easy one that I overlooked. I tried to prove the inequality but failed to find the right approach. I tried to argue via step functions, but this seems not to work. Maybe one needs to restrict to finite measures to make the above true. Any help is appreciated.
Take $f=\chi_A$. Since $\chi_A^2 = χ_A$, your inequality reads $$ \mu(A) ≤ \mu(A)^2 $$ so that $\mu(A)\geq 1$. This is false if $0<\mu(A)<1$.
Remark: This is actually true if $\mu(A)\geq 1$ since in that case $$ ∫_\mu \chi_A\,f ≤ ∫_\mu |f| = \|f\|_{L^1(\mu)} ≤ \mu(A) \,\|f\|_{L^1(\mu)} $$