I am going over a tutorial in my real analysis course. There is
an proof in which I don’t understand some parts of it.The proof relates to the following proposition:
(S – infinite σalgebra on X) ⟹ S is uncountable.
Proof:
Assume: S={Ai}+∞i=1. ∀x∈X:Bx:=∩x∈AiAi. [Note: Bx∈S ⟸ (Bx – countable intersection].
Lemma: Bx∩By≠∅⟹Bx=By.
Proof(of lemma):
z∈Bx∩By⟹Bz⊆Bx∩By.
1.x∉Bz⟹x∈Bx∖Bz∧Bx∖Bz⊂S∧Bx∖Bz⊂Bx
(contradiction: definition of Bx) ⟹ Bz=Bx2.y∉Bz⟹y∈By∖Bz ∧ By∖Bz⊂S ∧ By∖Bz⊂By(contradiction: definition of By) ⟹ Bz=By ⟹Bx=By ◻
Consider: {Bx}x∈X. If: there are finite sets of the
form Bx then: S is a union of a finite number of disjoint sets
⟹ S is finite ⟹ there is an infinite number of sets of
the form Bx. ⟹ ⋃i∈A⊆NBxi≥ℵ0.(contradiction) ◻There are couple of things I don’t understand in this proof:
Why the fact that we found a set (Bx∖Bz) in S
containing x and is strictly contained in Bx a contradiction
?Why if there are only a finite number of different sets of the
form Bx then S is a union of a finite number of disjoint
sets and is finite ?
Answer

Because Bx is supposed to be the intersection of all measurable sets containing x, but you’ve found a measurable set containing x strictly inside Bx.

Because for any measurable set T, we have T=⋃x∈TBx. Thus, if there are n distinct sets of the form Bx, then there are at most 2n elements of S.
Attribution
Source : Link , Question Author : Belgi , Answer Author : Zev Chonoles