Get the answer to "One of the "Remarks" in our lecture notes..."

Augustus Acevedo

Answered question

2022-07-02

One of the "Remarks" in our lecture notes says that if

G

is non-empty family of sets and

σ_{R} (G)

is the sigma ring generated by

G

, then every set in

σ_{R} (G)

is a subset of countable union of sets in

G

. Why is this remark true.

Janiyah Patton

Beginner2022-07-03Added 12 answers

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Let

X := {A : A is contained in a countable union of sets in G}

Observe that
1.

\emptyset \in X

X

is closed under countable unions.
3.

X

is closed under differences (in fact, if

B \subset A \in X

, then

B \in X

).
4.

G \subset X

Do you see how to complete the argument?

Waldronjw

Beginner2022-07-04Added 2 answers

X

is a

σ

-ring containing

G

, hence

σ_{R} (G) \subset X

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