$\sigma$-algebra I have a question about $\sigma$-algebra.
Let $(S,\Sigma)$ be a measurable space. Let $A \in \Sigma$. We can define $A \cap \Sigma:=\{A \cap M:M \in \Sigma\}$ and $A \cap \Sigma$ is $\sigma$-algebra on $A$. 
Question
Can we claim $A \cap \Sigma \subset \Sigma$ ? Is this meaningless?
 A: Yes, that statement makes sense. I would like to invent a different notation which I hope will be clearer:
If $\langle S, \Sigma\rangle$ is a measurable space, and $A \in \Sigma$ is a measurable set, define
$$\Sigma_A \equiv \{A\cap M : M\in \Sigma\}.$$
What sort of object is $\Sigma_A$?  It is a collection of subsets of $S$, just like $\Sigma$ is a collection of subsets of $S$.
Because the intersection of any two measurable sets is measurable, it follows that each member of $\Sigma_A$ is in fact a measurable set (each member is the intersection of the measurable set $A$ and some other measurable set $M$.)
Because $\Sigma$ contains all the measureable sets, we know that $$\Sigma_A \subseteq \Sigma.$$

I think one way to clear up the confusion is to keep careful track of which sets contain points (like $S$ and $A$) and which objects contain subsets of $S$ (like $\Sigma$ and $\Sigma_A$).
As whether you need to include the inclusion $i:A \hookrightarrow S$: we can either deal with these objects $S$, $\Sigma$, $A$, $\Sigma_A$ as sets, or we can remember that they are in fact the ingredients for particular measurable spaces.
If we are dealing with them as sets, then we can just write
$$\Sigma_A \subseteq \Sigma$$
which is a true statement because each element of $\Sigma_A$ is indeed an element of $\Sigma$.
But we could also notice that $\Sigma_A$ allows you to define a new measurable space — a measurable subspace of $\langle S, \Sigma\rangle$. This subspace is  $$\langle A, \Sigma_A\rangle$$
which you can prove satisfies all the axioms of a measurable space. And there is a measurable function (a morphism of measurable spaces) between them, given by the inclusion 
$$ i : \langle A, \Sigma_A\rangle \hookrightarrow \langle S, \Sigma\rangle$$
which sends points in $A$ to their equivalents in $S$. 
Moral: we only need inclusions when we are dealing with these objects as spaces rather than collections of points.
