# convergence of sets and limit of an integral

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: $m(x,h)=\int_{S(x,h)}f(x+h,y)dy$

where $S(x,h)= [ y \in Y:x \leq s(y) < x+h ]$ with $h>0$ and small.

Finally, $\forall(x,y)\in X\times Y$ such that $s(y)=x,f(x,y)=0$.

Question: Calculate the limit as $h\rightarrow0$ of $m(x,h)$

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Is this an exercise from a course? – Yemon Choi Jan 29 '13 at 22:12
Do you want to tell us why you're asking this question? (see the FAQ) – Anthony Quas Jan 29 '13 at 22:13
Sandy, this website is for questions of research interest. Is there a research angle to your question? If not, it might fit better at math.stackexchange.com (but if you do post it there, be sure to explain how you came upon the question, why it interests you, what you know about it, what progress you've made on it, where you get stuck, and so on). – Gerry Myerson Jan 29 '13 at 22:24
Yemon: it is not an exercise. I am a phd student in quantitative sociology and it is where a model that I wrote done led me... Anthony: I am bit rusty with convergence of sets and don't really know where to look for an answer. Gerry: this is of research interest to me. but this might be indeed a bit easy for mathematicians hihi. – Sandy Jan 29 '13 at 23:00
Mathematical questions from other research areas are usually well-accepted here, including questions with mathematical content of elementary level. However, for the sake of a profitable exchange, some explanation and motivation about the problem is welcome. – Pietro Majer Jan 30 '13 at 3:42