Infinite averages

Question

If you wanted to find the average of infinite items, what would it be?

Would an estimate of the first x averaged be a good estimate?

Or would the value be nearly zero because you are dividing by infinity?

I mean, basically the summation has to be infinity, but you're also dividing by infinity.

How do I solve this problem?

Answer 1

(I'm going to assume you mean arithmetic average, but the same applies for others too.)

This doesn't generally make any sense for several reasons, mainly because you can't look up an infinite amount of values to calculate the average of.

We can take the average of the first $n$ values and let $n$ grow, but this means that the order of the values in this "infinite list" is going to affect the average value you've calculated.

The only way this can work is if 1) We can find some sort of pattern in the values and 2) The value

$$\lim_{n \to \infty}\frac{x_1 + x_2 + \dots + x_n}{n} \to 0$$

Then the average will be zero. An example would be if $x_i = 1/i^3$ for all $i=1,2,3,\dots$

Answer 2

You need to endow your infinite set with a measure such that the whole space has measure $1$ and then integrate (and hope that your function is measurable to begin with). For finite sets, the obvious choice of measure is the counting measure divided by the number of elements; this gives the usual average. Even weighted averages can be achieved by playing with the measure. However for infinite sets, things are more difficult than just counting ...