# Question on the convergence $⨍_{\{f(x,y+s)>0\}} \to ⨍_{\{f(x,s)>0\}}$ as $y\to 0$

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!

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