So I was playing around with the notion of a functional integral. Basically given a set $S$ of functions we can define

$$ \int_{f \in S} L(f) $$

As the sum of of every function $f$ evaluated by the operator L. Such as

$$ \int_{f \in S} f' $$

One naturally issue of course that arises here, is how, to go about defining a sum over every function.

For the riemann integral if we assume the set we are working over is a convex line segment (so an integral of a single real variable)

$$ \int_{x \in S} f(x) $$

one can look at $C_1 = \{\inf(S), \sup(S)\} $ and define $C_{k+1} = \{\frac{x+y}{2} s.t. x \in C_{k}, y \in C_{k}\} $

Then define $$ \int_{x \in S} f(x)_{C_r} = \sum_{u\in C_r} \left[ \frac{1}{2^r+1} f(u) \right]$$

Then $$ \int_{x \in S} f(x) = \lim_{r \rightarrow \infty} \int_{x \in S} f(x)_{C_r} $$

The key here is that an infinite sequence of sets $C_r$ is created such that for any element in the original set $E \in S$ and for any $T > 0$ there will exist a natural number R such that there exists an element $u \in C_{r > R} $ such that $|E- u| < T$.

So now here is the the question. Given a set of functions S, that we assume is convex. so if $f,g \in \lambda f +(1- \lambda)g \in S (\lambda \in [0,1]) $

How does one create an infinite sequence of countable sets $C_r$ such that for any function $f \in S$ and for any $T >0$ there exists a natural number R such that there exists a function $u \in C_{r > R}$ such that $|\int_{x\in D}{[u(x) -f(x)]}| < T$?

Assume all functions $m \in S$ are defined over a domain $D$

The ability to generate such a scheme to create a countable set of functions that arbitrarily "approximates" any function from some target set $S$ is I think crucial in giving a rigorous definition to the idea of a "functional integral"


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