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.

• Relevant related question on MO: mathoverflow.net/questions/1388 – mrf Apr 22 '16 at 8:56
• Possibly functional integrals or (in QM) path integrals. – Raymond Manzoni Apr 22 '16 at 9:00
• @ Raymond: wonderful answer. My question precisely comes from some seemingly nasty functional integrals arising from a collision between dynamical system theory and Bayesian probability theory. Please check MO questions: mathoverflow.net/questions/232043/… , mathoverflow.net/questions/236527/… and mathoverflow.net/questions/236619/…. – Fabrice Pautot Apr 22 '16 at 9:12
• If somebody told me about summing $\mathfrak c$-many elements, I probably would not think of integrals first. There is also this type of sum (see also related questions) which make sense for any cardinality of the index set. – Martin Sleziak Apr 25 '16 at 6:46
• With measure theory you can integrate in every measurable space, despite its cardinality. As an example you can consider $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$ where $\mu$ is the only measure such that $\mu({n})=1\, \forall n \in \mathbb{N}.$ In this case integration formally reduces to series in a very natural way. This means that integral is not the way to sum over cardinality $2^\omega$ but rather a way to generalize sums to every cardinal. If you are looking for a concrete measure or $\sigma$-algebra structure on the set $\mathbb{R}^\mathbb{R}$, well that's another question. – user459312 Jul 3 '17 at 5:36