As the title says, I have a function, which vanishes for increasing time almost everywhere. For example take the function
$$ \begin{equation} \label{markov} f(t) = \left\{ \begin{array}{ll} \frac{1}{t} &\text{for } t \in\mathbb{R}\setminus\mathbb{N}, \\ 1 &\text{else}. \end{array} \right. \end{equation}$$ Now, for increasing time, $f$ goes to $0$ for almost all $t$, but for natural arguments $f$ is constant $1$. I guess i cannot write $$\lim_{t\rightarrow\infty}f(t) = 0 \quad \text{almost everywhere.}$$ But how can i write this?
Note that one usually defines almost everywhere convergence for a sequence of functions $(f_n)_{n\in\mathbb{N}}$. But I think we can get what you want by imitating the notion of $\liminf $ and $\limsup$.
In the comment by David C. Ullrich, one has to know the negligible subset preventing $f$ from converging. Without this data a priori, one could look at $$ \lim_{t\to\infty} \big( \underset{s\geq t}{\operatorname{ess inf}} f(s)\big)\quad \text{and}\quad \lim_{t\to\infty} \big( \underset{s\geq t}{\operatorname{ess sup}} f(s)\big)$$ If they coincide, then this value is the limit, otherwise there is no limit.
The essential infimum or supremum (here on $\mathbb{R}$) are defined by $$\underset{s\in \mathbb{R}}{\operatorname{ess inf}} f(s) := \sup \big\lbrace m\in \mathbb{R},\ f^{-1}\big(]-\infty,m[ \big)\ \text{negligible}\big\rbrace \tag{1}\label{1}$$ $$\underset{s\in \mathbb{R}}{\operatorname{ess sup}} f(s) := \inf\big\lbrace M\in \mathbb{R},\ f^{-1}\big(]M,+\infty[ \big)\ \text{negligible}\big\rbrace \tag{2}\label{2}$$ (In comparison, the infimum is the "greatest lower bound" and the supremum is the "least upper bound", i.e. $$\inf_{s\in \mathbb{R}} f(s) := \max\big\lbrace m\in \mathbb{R},\ f^{-1}\big(]-\infty,m[ \big)=\emptyset\big\rbrace$$ $$\sup_{s\in \mathbb{R}} f(s) := \min\big\lbrace M\in \mathbb{R},\ f^{-1}\big(]M,+\infty[ \big)=\emptyset\big\rbrace$$ I'm not exactly sure if one really needs $\inf$ and $\sup$ in (\ref{1},\ref{2}) or if one could put $\min$ and $\max$, but this is left to the reader...)