$\mathcal{L}^1$-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?

For a simple positive function $f$ that takes values $c_1,\ldots,c_n$ on sets $E_1,\ldots,E_n \in \Sigma$, by definition
$$\mu(f 1_E)=\sum_{i=1}^n c_i \mu(E \cap C_i) = \mu_E(f_E).$$
For an arbitrary positive $\Sigma$-measurable function $f$,
$$\mu(f 1_E)=\sup\{ \mu(g) \mid g \in \Gamma_1 \}$$
$$\mu_E(f_E)=\sup\{ \mu_E(g) \mid g \in \Gamma_2 \}$$
where $\Gamma_1$ and $\Gamma_2$ are sets of all $\Sigma$ and $\Sigma_E$ measurable simple positive functions $g_1$ and $g_2$ such that $g_1(s) \leq f 1_E(s)$ and $g_2(s) \leq f(s)$ for all $s$ in $S$ and $E$ respectively. But, it is easy to show that $$\Gamma_2 = \{ g_E \mid g \in \Gamma_1 \}.$$
