convergence of sets and limit of an integral - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:51:52Z http://mathoverflow.net/feeds/question/120263 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120263/convergence-of-sets-and-limit-of-an-integral convergence of sets and limit of an integral Sandy 2013-01-29T21:50:40Z 2013-01-29T21:50:40Z <p>Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets. Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function. Let $s:Y\rightarrow X$ be a function (not necessarily continuous). Define $m:X\times\mathbb{R}\rightarrow\mathbb{R}$ as: $m(x,h)=\int_{S(x,h)}f(x+h,y)dy$</p> <p>where $S(x,h)= [ y \in Y:x \leq s(y) &lt; x+h ]$ with $h>0$ and small.</p> <p>Finally, $\forall(x,y)\in X\times Y$ such that $s(y)=x,f(x,y)=0$.</p> <p>Question: Calculate the limit as $h\rightarrow0$ of $m(x,h)$</p>