In my course, we have defined pullback and pushforward as follows. Let $f:X\to Y$ with $A\subseteq X$ and $B\subseteq Y$. Then the pullback $f^*$ of a collection of subsets $\{B_1,B_2,\cdots\}$ is defined $f^*\{B_i \}=\{f^{-1}(B_i)|B_i\in \{B_i \} \}$. Similarly, the pushforward we defined $f_*\{A_i \}=\{\{B_i\} \subseteq Y|f^{-1}B_i \in \{A_i\} \}$. Now, we have used these concepts to show collections are sigma algebras. In particular, we said that if $X\overset{f}{\to}(Y,\Sigma)$, then $f^*\Sigma$ is a sigma algebra on $X$.
I am considering a set $X$ equipped with a sigma algebra $\Sigma$ and I want to show that the collection $\Sigma_A = \{E\in\Sigma|A\subset E\text{ or }A\cap E=\emptyset \}$ is a sigma algebra on $X$. My approach, using this method of pullback and pushforward (as the professor likes) is as follows.
Consider the canonical injection $\iota: A\hookrightarrow X$. Since $\Sigma$ is a sigma algebra on $X$, we notice that $\iota^*\Sigma=\{A\cap E|E\in\Sigma \}$ is a sigma algebra on $A$. But since we want a sigma algebra on $X$, we take the pushforward of this: $\iota_*\iota^*\Sigma=\iota_*\{A\cap E|E\in\Sigma \}=\{B\subseteq X|\iota^{-1}(B)=A\cap B\in \{A\cap E|E\in\Sigma \} \}$. So $B$ must be a $\Sigma$ measurable set. I don't really know how to proceed from here. In particular, I don't know how to get $A\subseteq E$ or $A\cap E=\varnothing$ from tis.