Timeline for Echange of Infimum Integral with Pointwise Infimum

9 events

when toggle format	what		by	license	comment
Oct 23, 2018 at 1:21	vote	accept	ABIM
Oct 16, 2018 at 12:22	comment	added	Dirk		Just noticed something strange: If $g$ in your question is convex and $f$ is real valued, then $g(f(x))\leq M$ just says that $f(x)$ is in some closed interval. Is it that what you mean? If yes, then the infimum is indeed attained pointwise and $f^2(x)$ should be equal to the smallest possible value everywhere.
Oct 16, 2018 at 12:17	comment	added	Dirk		If would be helpful if you could assure that the integrand on the right hand side, i.e. the function $g(x) = \inf_{f\in U} f^2(x)$, is in $U$, in other words, that $U$ is closed under taking infima. If not, I would suspect that the equality is not true in general.
Oct 16, 2018 at 11:52	answer	added	Martin Kell		timeline score: 1
S Oct 13, 2018 at 16:43	history	suggested	Amir Sagiv	CC BY-SA 4.0	relevant subject tag+ abbreviation is little known, so I "expanded" it
Oct 13, 2018 at 16:30	comment	added	ABIM		That's interesting....I'll give that some though, at the moment I'm looking at this book: sites.math.washington.edu/~rtr/papers/… By Rockafellar and Wets. The last chapter may be able to help :)
Oct 13, 2018 at 16:09	review	Suggested edits
S Oct 13, 2018 at 16:43
Oct 13, 2018 at 16:01	comment	added	Nate Eldredge		The usual approach would be to try to find a sequence $f_n$ converging a.e. to the infimum, and then try to use something like the dominated convergence theorem on the functions $f_n^2$.
Oct 13, 2018 at 15:28	history	asked	ABIM	CC BY-SA 4.0