Mean value of a function over $R$

Question

Over an interval $[a, b] $ we can define the mean value of a function $f$ as $$\overline{f} = \frac{\int_{a} ^{b} f(x) dx} {b-a}. $$ Is there a formula for the mean value of a function over $R$?

Answer 1

One way of extending it would be to take the limit symmetrically: $$\lim_{c \to +\infty}\frac{1}{2c}\int_{-c}^c f(x)\,\mbox{d}x$$ Of course, functions with a finite integral on $\mathbb{R}$ will then have a mean of $0$. Constant functions have a mean equal to their function value.

For example for the function $e^{-a|t|}$ the outcome with this formula is zero. It doesn't make any sense.

The mean of $e^{-|kt|}$ on $[a,b]$ becomes arbitrarily small if you take $b-a$ sufficiently large. Why would any finite but strictly positive number make (more) sense to you, in this case?