Let $f$ be a continuous function in both variables, i.e $f \in C(U \times [0,M])$ for some $M< \infty$ and $U$ compact subset of $\mathbb R^2$

QUESTION: If $\vert \{f(x,s)>0\} \vert <\infty$, is continuity in time sufficient in order to claim the following convergence: $⨍_{\{f(x,y+s)>0\}} \to ⨍_{\{f(x,s)>0\}}$ as $y\to 0$

where $⨍$ denotes the average integral over the corresponding sets.

I believe that the answer is positive. However, since the average integral operator is involved, I 'm getting a bit puzzled about how trivial my question is.

Any help is much appreciated. Thanks in advance!

  • $\begingroup$ Is $\{f(x,y)>0\}$ of finite measure? Otherwise I wonder how you define the average. $\endgroup$ – amsmath Oct 9 at 16:51
  • $\begingroup$ @amsmath yes, it is. I will add it $\endgroup$ – kaithkolesidou Oct 9 at 16:52
  • $\begingroup$ @amsmath thanks. stupid typo $\endgroup$ – kaithkolesidou Oct 9 at 16:55
  • $\begingroup$ Note that $\frac{1}{|A|}\int_A\,dx = 1$. What do you mean by the average integral? Please write it out. $\endgroup$ – amsmath Oct 9 at 16:59
  • $\begingroup$ @amsmath For the average integral I use the same definition as you do... So, the convergence is trivial? $\endgroup$ – kaithkolesidou Oct 9 at 17:03

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