I have a positive function $f(t,x,y)$ ($0\leq f(t,x,y)\leq 1$ for all $x,y\in\mathbb{R}$, $t\in[0,\infty)$) . Let us further assume that $\lim_{t\rightarrow\infty}f(t,x,y)$ does not exist. Now, let $\mu$ be a finite absolutely continuous measure on $\mathbb{R^2}$. Can the $L_{1}(\mathbb{R^{2}},\mu)$ limit of $f(t,x,y)$ exist? If so, can it ever be zero?
I know that $L_{1}$ convergence implies that there is a subsequence $\{f(t_{k},x,y)\}$ which convergence pointwise a.e. This is what I was trying to use; however, because the set of functions $\{f(t,x,y)\}_{t}$ is bounded, this is not helpful since bounded subsequences will exist in such a case. I think that there must be some simple theorem that sheds light on this that I am missing. Any aid would be greatly appreciated.
Yes, let $S_n$ be sequence of sets s.t. every point of $\mathbb R^2$ is covered by infinitely many of them, and also not covered by infinitely many of them, but $\mu(S_n) \to 0$ - for example, let $A_i^j$, $j = \overline{1,4i^4}$ be coverage of square $[-n, n] \times [-n, n]$ by squares with side $\frac{1}{n}$, and $S_n$ be sequence of all $A_i^j$, ordered by $i$ first, $j$ second.
Then let $f(t, x, y) = \begin{cases} 1&& (x, y) \in S_{\lceil t\rceil} \\0\end{cases}$. $\lim\limits_{t\to \infty} f(t, x, y)$ exists nowhere, because $f(t, x, y)$ can be $0$ or $1$ for arbitrary large $t$. But $\int_{\mathbb R^2} f(t, x, y)\, dx\,dy = \mu(S_{\lceil t\rceil}) \to_t 0$.