Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to prove some property about essential suprema and I think I can show them for the pointwise supremum $\sup S$.

The problem is, that the sets involved are uncountable and thus, the point-wise suprema need not be measurable.

Is there a way to transfer the results from the point-wise to the essential supremum?

I know the following does not make sense, since $\sup S$ might not be measurable, but is there something comparable?

$\mathrm{ess}\sup S=\sup S \;\;$ almost surely


If it helps, the statement I would could prove for an uncountable collectino of uncountable sets of random functions $\{S_i\}_{i\in I}$:

$\inf \bigcap_{i\in I}S_i = \sup_{i\in I}\left(\inf S_i\right)$

Now I want to show

$\mathrm{ess}\inf \bigcap_{i\in I}S_i = \mathrm{ess}\sup_{i\in I}\left(\mathrm{ess}\inf S_i\right)$


I cross posted to Mathoverflow

share|improve this question
    
What is the essential supremum of a set? –  Michael Greinecker Feb 15 '13 at 18:55
    
it is defined for a set of real-valued measurable function. (see planetmath.org/EssentialSupremum.html) –  user4514 Feb 15 '13 at 19:25
    
So why are you taking intersections? –  Michael Greinecker Feb 15 '13 at 19:27
    
The intersection of a collection of sets of functions. Sorry, if it is not clear. –  user4514 Feb 16 '13 at 10:07

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.