# 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: ???

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. 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/…. 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. 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. Jul 3 '17 at 5:36