Hello all, I am trying to prove that a countable union of null sets is null, i.e.

is null when each

is null. Now I am a bit confused about the notion of an infinite union. Is it enough to show that

is null for all

?

I have managed to do that in the following way: since you pick

, then for

these are all null so you can define sequences

with

. Hence we define a sequence

by putting all the sequence elements of

up to

in the sequence, ie the sequence

. Of course

, and we have that

. Therefore

is null for all

.

But I am confused about the infinite union? Can I deduce it from the above?

what about choosing

, then for each

, choose a sequence

such that

Then I would have to define a sequence

which somehow contains infinitely many of the above sequences, so that I get

. But can I construct a sequence like that?

Any help would be appreciated, thank you