$\mathcal{L}^1$-integrability with restriction to a set $E$ Let $E \in \Sigma$. We look at the measurable space $(E,\Sigma_E)$, where $\Sigma_E = \{E \cap F: F \in \Sigma\}$, which is a $\sigma$-algebra. We consider integration to this space.
Let $f$ be a measurable function and denote by $f_E$ its restriction to $E$. Now, I want to prove by standard machinery that
\begin{align}
f_E \in \mathcal{L}^1(E,\Sigma_E,\mu_E) \iff \mathbb{1}_E\ f \in \mathcal{L}^1(S,\Sigma,\mu),
\end{align}
in which case the identity $\mu_E(f_E) = \mu(\mathbb{1}_E\ f):=\int_E f\ d\mu$ holds.
Now, I am wondering if we can use the fact that $f_E = f\ \mathbb{1}_E$ in the elaboration of the standard machinery. If this is the case, the results follows easily. Otherwise, I do not know how to prove the above identity. Any suggestions?

Note that $f_E$ is NOT equal to $f1_E$. Simply because domains of $f_E$ and $f1_E$ are different: the former is $E$ and the latter if $S$.
However, by definition, $\int_E fd\mu_E=\int_S f1_Ed\mu$ and this is something you can always use.
