We have sums, series and integrals. What's next?

We know how to sum or average a finite number of terms: sums.

We know how to sum a countable infinite number ${\beth_0}$ of terms: series.

We know how to sum ${\beth_1}$ terms: integrals.

How to sum ${\beth_2}$ terms: ???

One "concrete" example please.

Let ${\mathbb{R}^\mathbb{R}}$ be the set of all functions $f:\mathbb{R} \to \mathbb{R}$.

Let ${x_0} \in \mathbb{R}$. Let ${\mathbb{R}^\mathbb{R}}\left( {{x_0}} \right)$ be the subset of all functions in ${\mathbb{R}^\mathbb{R}}$ having ${{x_0}}$ in their domains of definition. There are still ${\beth_2}$ of them.

Is the "functional mean image" of ${x_0}$ under all functions in ${\mathbb{R}^\mathbb{R}}\left( {{x_0}} \right)$ that we can formally write as

$\int\limits_{{\mathbb{R}^\mathbb{R}}\left( {{x_0}} \right)} {{\text{D}}f\,f\left( {{x_0}} \right)\,\,} $

(well) defined? If not, why?

Same question in the set of all bijections from $\mathbb{R}$ to $\mathbb{R}$.

This hypothetical "functional mean image", to be compared to the usual ${\beth_1}$ integral

$\int\limits_\mathbb{R} {{\text{d}}xf\left( x \right)} $

• may be (well) defined in some branch of mathematics I (or you) do not know;
• may be an unidentified mathematical object;
• may not exist.

Anything welcome. My apologies if it is trivial but I sincerely do not know. Thanks.

Path integrals in quantum mechanics take averages of all the ways a particle could move from one point to another.