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Normally in probability theory, you have a Random Variable $X$ that lies in the probably space $(A,\Sigma, P)$, with distribution function $F_X(x) = P\{X^{-1}(-\infty,x]\}$.

What happens when you have a random function $f$, that is assigned a function at random in some probability space? How do you formalise its distribution function?

NB: I've had a look at wikipedia but I don't find it rigorous enough to truly understand its formulation. It would be great to know if anyone had any rigorous understanding of the mechanisms of a random function.

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Only real-valued ($\mathbb{R}^n$) random variables have what we call a distribution function. In case you have a random variable (or element, to signify that it is not real valued) with values in a measurable space $(S,\mathcal{S})$, defined on a probability space $(\Omega,\mathcal{F},P)$, we can only really talk about its distribution.

As per definition we say that the distribution of a random element (by definition a measurable mapping from a probability space to a measurable space) $X:(\Omega,\mathcal{F},P) \to (S,\mathcal{S})$, is given by the pushforward measure $X(P)\equiv P\circ X^{-1}$ on $(S,\mathcal{S})$. Now for any measurable set $A\in \mathcal{S}$ we have that $P(X\in A)=P(X^{-1}(A))=X(P)(A)$.

If none of the above made any sense read, i suggest studying some measure theory, followed by measure theoretic probability (e.g.).

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John
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