Sequences of Functions

Question 1

lim sup an exists if…

Answer 1

an is a bounded sequence

Question 2

f_n converges almost uniformly on X if …

(each f_n: X -> R)

Answer 2

fn –> f uniformly

on X - J

where for any epsilon,

there is a measurable J such that

u(J) < epsilon

Question 3

For a sequence f_n of measurable functions,

If f_n –> f almost everywhere, …

If f_n is bounded,

Answer 3

then f is measurable

then lim sup fn (and inf) are measurable!