I know this question is very similar to many asked but I have looked and could not find an exact match.
I have an integral formula which I know to be true, for all $\epsilon \in [0,h)$ for some $h \in \mathbb{R}$:
$$\int_Kf_{\epsilon}(x)d\mu(x) = Pf_\epsilon(k)$$
Where:
- $P$ is some constant,
- $\{f_\epsilon\}_{\epsilon \in [0,h)}$ is some set of functions (all defined on the same domain, denote the domain $D$)
- $k$ is some constant in $D$
- $K$ is some (not necessarily compact) subset of $D$
- $\mu$ is some measure.
I also know that $f_\epsilon(k)$ is continuous with respect to $\epsilon$ over $(0,h)$.
Define $\hat{f}(y)$ by; for all $y \in D$:
$$\hat{f}(y) = \lim_{\epsilon \to 0} f_\epsilon(y)$$
$\hat{f}(k)$ might not equal $f_0(k)$ but is always finite. My question is; does it follow that:
$$\int_K\hat{f}(x)d\mu(x) = P\hat{f}(k)$$
And if not, under what conditions does it fail? Would anyone be able to give me an example?
Thank you in advance!