I am trying to self learn from a MIT-OCW course on measure theoretic probability: this resource. However, one of their homework problems turns out to be very messy even in the solution provided (which is probably why it is optional).
In their homework assignment 2, question 7 the following shows up in their proof: The very first part is where I have trouble,
For a collection $C$ and a space $Ω$ let $α_Ω (C)$ denote the smallest algebra of sets in $Ω$ containing $C$. Here $Ω_1 \subset Ω$
Claim: $α_{Ω_1} (C ∩ Ω_1) = α_Ω(C) ∩ Ω_1$ .
I understand the intent to show the 2 are equal, by showing they are contained in one another but in doing so they arrive at the following step: let $E∩Ω_1 ∈ α_Ω (C)∩Ω_1$ , then
$$
(E ∩ Ω_1)^c= Ω_1 -(E ∩ Ω_1 ) = E^c ∩ Ω_1 ∈ α_Ω (C) ∩ Ω_1\text{, as } E ∈ α_Ω (C)
$$ and $ α_Ω(C)$
is an algebra.
As I understand , $(E ∩ Ω_1)^c$ = $ Ω_1 -(E ∩ Ω_1 )$ can only happen when $E \subset Ω_1$ holds, but this is not the general case. I am stuck here and cannot process rest of the proof, now I have spent 2 days trying to convince myself that the steps are right.
Can someone provide some clarity? The proof is actually very badly written, with an attempt to squeeze in some general theorem about algebras and their intersection.
It's a simple overload of notation:
Since we want to prove that $\alpha_\Omega(C) \cap\Omega_1$ is an algebra of sets over $\Omega_1$, we use the complement operation relative to $\Omega_1$ in $(E\cap\Omega_1)^c$.
(Note, on the other hand, $E^c$ in the same line refers to complement relative to $\Omega$.)
Then they will want to prove that it's the smallest such set algebra. So this is not exactly the standard method of proving equality of two sets by two way inclusions, instead it directly uses the cited definition.
By the way, one can think about it in a more constructive way: each element of $\alpha_X(Y)$ (with any $X$ and $Y\subseteq P(X)$) can be written up by repeatedly using unions and complements on elements of $Y$.
Now to prove the claim, just simultaneously perform the same operations on the elements of $C$ and $C\cap\Omega_1$ (by which really $\{H\cap\Omega_1:H\in C\}$ is meant here).