Re: Egoroff's Theorems Proof

1) We write the set on which converges pointwise to f as an union of such sets. We take instead of in order to get a countable family.

2) Since for any m, we use the definition of the limit to to choose such that (we can choose it to have the inequality for , but it's not necessary here).

3) We used the fact that when we wrote .

Re: Egoroff's Theorems Proof

Quote:

Originally Posted by

**girdav** 1) We write the set on which

converges pointwise to f as an union of such sets. We take

instead of

in order to get a countable family.

2) Since

for any m, we use the definition of the limit to

to choose

such that

(we can choose it to have the inequality

for

, but it's not necessary here).

3) We used the fact that m(A)<\infty when we wrote

.

Can the set on which converges pointwise to f be written as { } ?

Re: Egoroff's Theorems Proof

Yes, you can see that using the definition of the limit.

Re: Egoroff's Theorems Proof