Suppose we have a 'random function' $f(x)$ that will be one of $n$ functions $f_1(x), ..., f_n(x)$ with probabilities $p_1, ..., p_n$. Intuitively, it might seem sensible to define
$$E[f(x)] = \sum_{i=1}^{n}f_i(x)p_i$$
Now suppose $f(x)$ can be any one of a continuum of functions, each indexed by a unique real number. Again, it might seem sensible to define
$$E[f(x)] = \int_{-\infty}^{\infty}f_i(x)g(i)di$$
where $g(i)$ describes the probability densities of the different possible functions.
However, I cannot find such expressions anywhere; and when I Google 'functional integration' I find rather more complicated notions than those expressed here.
Is there some reason why the definitions above are invalid/problematic? To be clear, I understand that the set of real valued functions is larger than the set of real numbers. So if all real valued functions had positive probability, it would not be possible to index them with the real numbers as I have done here. However, suppose we somehow knew that there were only a continuum of functions with positive probability: could we then define the expected value of the function as above?
Many thanks in advance!