We know that the average of a function $f(x)$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ in the interval $[a,b]$ can be computed by $$ \langle f(x) \rangle =\frac{1}{b-a}\int_a^b f(x)dx. $$ How should we define the average of a function in an infinite domain, say $[a,+\infty)$ and $(-\infty, \infty)$, given that the limits $\lim_{x \rightarrow \pm \infty} f(x)$ exist? And how do we extend this to functions $f: \mathbb{R}^n \rightarrow \mathbb{R}$? Is it, for example, reasonable to say that the average of a function $f(\vec{x})$ is $$ \langle f(\vec{x}) \rangle = \lim_{L \rightarrow \infty} \frac{1}{L^n} \int_{-L}^L \int_{-L}^L \dots \int_{-L}^L f(\vec{x})d\vec{x} $$ ? What about choosing another coordinate system, say, a generalized spherical coordinate system instead, and integrate the function in a high-dimensional sphere with the radius approaching infinity? What properties does the function have to fulfill to ensure that all coordinate representations give equal results?
If any illustrative examples are used, I would prefer that a high-dimensional Gaussion function could be taken as an example, which is a perfect example of function that approaches to zero in all directions.
The problem that I am originally dealing with is to calculate the variance of the linear combination of Gaussian functions centered at different locations, which involves evaluating the average of something like this $$ \sum_{i,j} e^{-[(\vec{x}-\vec{x}_i)^2 + (\vec{x}-\vec{x}_j)^2]/2\sigma^2} $$ over the entire $\mathbb{R}^n$ space.
The answer to your question depends on what you mean by "average". If you want an operation on some functional space then you have a lot of possibilities, even if you ask some properties as linearity, continuity etc. The more general construction to my knowledge is
$$ \langle f \rangle = \int_X f d\mathbb P $$ which is the expectation of $f$ seen as a random variable on a probability space $(X, \mathcal T, \mathbb P)$. What you call "the" average of $f$ is the last construction with the uniform probability on $[a,b]$. It seems natural because it is the only probability measure on $[a,b]$ that is a multiple of the Lebesgue measure, which itself satisfies an intuitive property, but you might want to weight differently your space, for instance with a normal distribution.
Considering your original question, you want to specify your sample of Gaussian variables because the law of the sum of Gaussian variables can vary a lot given the cases. For instance it is not even sure it is Gaussian, so the theoretical computations might be complicated.