# Prove limit inferior (lim inf) is measurable?

• May 13th 2010, 04:37 PM
rodl
Prove limit inferior (lim inf) is measurable?
f_n measurable

Prove lim inf f_n is measurable?

There's supposedly a proof in Royden's Real Analysis, but unfortunately I do not have access to it.

Anyone?

Much appreciated.
• May 14th 2010, 05:03 AM
Idealconvergence
for all a in R, {x in E: limsup{f_n: n in N}(x)>a},
={x in E: limsup{f_n(x): n in N}>a}
= Union {x in E: f_n(x) > a}
= a measurable set
• May 14th 2010, 05:13 AM
rodl
Quote:

Originally Posted by Idealconvergence
for all a in R, {x in E: limsup{f_n: n in N}(x)>a},
={x in E: limsup{f_n(x): n in N}>a}
= Union {x in E: f_n(x) > a}
= a measurable set

Thank you. But that's for lim sup (I already knew that proof). For lim inf I can't just take the intersection as it messes up difference between >= and >. I think I have to go about by investigating the complements but I'm not sure.
• May 14th 2010, 08:44 AM
rodl
omg.

lim inf f_n = -lim sup (-f_n)

(Clapping)