signed measure and expected value Let $(\Omega, F, \mathbb{P} )$ be a probability space. Let $\mathbb{G}$ be a sub-algebra of $F$ . 
If $X: \Omega \rightarrow \mathbb{R}$ is a random variable with $\mathbb{E}[|X|] < \infty$. 
If $\mu(G) = \mathbb{P}(G), \nu(G) = \mathbb{E}[XI_G], G \in \mathbb{G}$. We need to prove that $\nu$ is a signed measure. 
I have problems with proving why there do not exist $G, H \in \mathbb{G}$ so that $\nu(G) = - \infty$ and $\nu(H) = \infty$.  Can someone help me?

Hint: $$|\nu(G)| \leq \mathbb{E}(|X|)<\infty$$ for any $G \in \mathbb{G}$.
Remark: Note that $\mathbb{E}(|X|)<\infty$ ensures that $\nu$ is well-defined. If $\mathbb{E}(|X|)=\infty$, then there exist $G \in \mathbb{G}$ such that $X \cdot 1_G$ is not integrable, i.e. the expectation $\mathbb{E}(X \cdot 1_G)$ is not defined.
