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mona123

Invité

integration

hi please i didn't know how to solve this problem .can someone help me :
Let f be a measurable function on a measurable set E with positive measure such that
f is finite a.e. Show that there exists a subset of E with positive measure on which f is
bounded.
thanks in advance.

Fred

Administrateur

Inscription : 26-09-2005

Messages : 7 352

Re : integration

Hi,

  Take [tex]E_n=\{x\in E;\ | f_n(x) |\leq M\}.[/tex]
Then the sequence [tex](E_n)[/tex] is increasing and the measure of [tex]\bigcup_n(E_n)[/tex] is positive...

Fred.

mona123

Invité

Re : integration

thank you Freddy but i realy didn't understand this problem .can you please write the solution more detailed.thanks again.

Fred

Administrateur

Inscription : 26-09-2005

Messages : 7 352

Re : integration

Because the measure of [tex]\cup_n E_n[/tex] is positive, there exists some [tex]n[/tex] such that the measure of [tex]E_n[/tex] is positive.
Thus, [tex]f[/tex] is bounded (by [tex]n[/tex]) on the set [tex]E_n[/tex] of positive measure.

F.