I am unfamiliar in statistics. My knowledge is in pure mathematics.
Suppose $n\in\mathbb{N}$, where $X$ is in the $\sigma$-algebra of Caratheodory-measurable sets such that $X\subseteq\mathbb{R}^{n}$ and $f$ is a measurable function where $f:X\to\mathbb{R}$.
Edit: For some measure $\mu$ on measurable space $(X,\Sigma)$, the average w.r.t to $\mu$, where $0<\mu(X)<\infty$, is:
$$\text{avg}_f(X)=\frac{1}{\mu(X)}\int_{X}f(x) \; d\mu$$
However, in mathematics $\mu(X)$ need not equal $1$ but I wish to know, in detail, how to get the same result using a probability measure.
Question: How does one use statistics to find the mean of $f$, using the uniform probability measure on $X$ (explained here and here in pg.32-37)?
I assume that in statistics the expected value is the same as the mean, but in textbooks, we're shown the expected value applies to the probability density function or the cumulative distribution function.
Furthermore, when the Lebesgue measure is defined $\sigma$-algebra of Lebesgue-measurable sets, which satisfy the Caratheodory criterion, I heard it's not a good idea to use the mean of $f$ w.r.t the uniform probability measure when:
- $X$ is countably infinite (the uniform probability measure does not exist)
- $X$ has infinite or zero Lebesgue Measure
- $f$ is an infinite number of points that covers an infinite expanse of space in general.
(Optional): How do we find the mean of $f$ for these cases?