Let
- $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space
- $f\in\mathcal L^1(\mu)$ and $$\mathcal M:=\left\{A\in\mathcal A:\int_Af\:{\rm d}\mu\ge0\right\}$$
- $\mathcal E\subseteq\mathcal M$ be an algebra with $\sigma(\mathcal E)=\mathcal A$
I want to conclude that $$\int_Af\:{\rm d}\mu\ge0\;\;\;\text{for all }A\in\mathcal A\;.\tag1$$
Using the dominated convergence theorem, we should be able to show that $\mathcal M$ is a monotone class and hence $$\mathcal A=\sigma(\mathcal E)\subseteq\mathcal M\subseteq\mathcal A\tag2$$ by the monotone class theorem. Is there any mistake in my argumentation?
Moreover, I'm curious whether or not we're able to relax the assumption of being an algebra on $\mathcal E$. In particular, I've found a proof where the author is assuming that $\mathcal E$ is only closed under complement. I've asked another question for that.