${f_n}$ : seq of measurable functions with common domain E, show that each of the following functions is measurable: inf{$f_n$},sup{$f_n$}

Asked Oct 15 '16 at 10:27

Active Oct 15 '16 at 10:27

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For a sequence ${f_n}$ of measurable functions with common domain E, show that each of the following functions is measurable: inf{$f_n$},sup{$f_n$},lim inf{$f_n$},lim sup{$f_n$}

At first, I want to know what inf{$f_n$} means.

If {$f_n$} is increasing function, then I understand {$f_1$} is the infimum of sequence of function {$f_n$}.

But what if it doesn't? If two candidates have a crossed point, then what is infimum of {$f_n$}?

Secondly proving the statement : Since {$f_n$} is measurable on E, for any real number $c$,

{$x\in E |f_n(x) >c$}(Or $\gt$ can be $\lt,\le,\ge$) is measurable.

Then {$x\in E$| inf{$f_n$} $\ge c$} , since for any $n$,$f_n$ $\ge$ inf{$f_n$}

so, done. Is it right?

asked Oct 15 '16 at 10:27

JAEMTO

The functions are the pointwise (!) infimum, supremum etc. of the sequence. – Daniel Robert-Nicoud Oct 15 '16 at 10:29
@DanielRobert-Nicoud Then you mean "for any point in E?" – JAEMTO Oct 15 '16 at 10:31
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This question is repeated http://math.stackexchange.com/questions/1456350/the-supremum-and-infimum-of-a-sequence-of-measurable-functions-is-measurable – Joaquin San Oct 15 '16 at 11:20

${f_n}$ : seq of measurable functions with common domain E, show that each of the following functions is measurable: inf{$f_n$},sup{$f_n$}

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