Applications of infinite cardinalities in real analysis

Question

What are some topics in real analysis that make use of infinite cardinalities larger than that of the real numbers themselves, preferably [edit: but not necessarily] topics that are widely applied in scientific applications?

Edit: While there may be not any actual infinities in science, the notion of infinity plays an essential part in real analysis which is indeed widely applicable in science, e.g. in the notion of a limit, the Fundamental Theorem of Calculus and the Intermediate Value Theorem . I was just wondering if higher orders of infinities are similarly required.

Answer 1

The requirement "widely applied in scientific applications" is probably too high of a bar to get anything of interest, but I can think of several examples in real analysis where cardinalities higher than $2^{{\aleph}_0} = c$ are applied.

The standard argument that there exists a Lebesgue measurable set that isn't a Borel set is an example: The Cantor middle thirds set has $2^c$ many subsets, all of which have Lebesgue measure zero, but there are only $c$ many Borel sets.

A similar argument shows there exist Lebesgue measurable functions that are not Borel measurable: There are $2^c$ many characteristic functions of subsets of the Cantor middle thirds set, but there are only $c$ many Borel measurable functions.

Miroslav Chlebík, in this 1991 Proc. AMS paper, showed there exists $2^c$ many symmetrically continuous functions from the reals to the reals, and thus there exist symmetrically continuous functions that are not Borel measurable. When Chlebík's paper appeared, it had been a long unsolved question whether there even exists a symmetrically continuous function that isn't a Baire one function. See also the math StackExchange question Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?.

Because the union of interior of the unit disk in ${\mathbb R}^2$ with any subset of its boundary is a convex set, there exist $2^c$ many convex sets in ${\mathbb R}^{2}.$ Since there are only $c$ many Borel subsets of ${\mathbb R}^{2},$ it follows that there exist convex sets in ${\mathbb R}^2$ that are not Borel. (Note that this is so not true in ${\mathbb R}.)$

Answer 2

My modest understanding of real analysis leads me to believe that infinite cardinals have no role to play here. Indeed, the thrust of the subject's development was to replace the actual infinities of its original formulation with the potential infinities we use today in expressing the notion of convergence and limits.

Regarding scientific applications, the finite nature of our science seems to imply that any role infinity may take can only be the result of our mathematical idealization and abstraction.