measure theory for regular cardinals

Question

Measure theory is somewhat focused on the cardinal $\aleph_0$: First of all we have the usual $\sigma$-additivity, Polish (separable!) spaces such as $\mathbb{R}^n$, countable sequences and limits, countable recursive constructions. I know that this is a vague question, but is there a possibility to extend measure theory to an arbitrary regular cardinal $\kappa$? Thus in the definitons we replace $\aleph_0$ by $\kappa$ and try to imitate the theory? Perhaps we should also replace $\mathbb{R}$ by some other model because in $\mathbb{R}$ every convergent sum is supported on a countable index subset. Or is there already some research on it?

The reason for my question is not only just of curiosity. I want to understand in detail this "cardinality boundary" of measure theory.

Answer 1

One answer to your question is to remain within the realm of the continuum, and consider what happens when the Continuum Hypothesis fails. In this case, there are cardinals κ below the continuum c = |R|, and one naturally inquires whether these cardinals behave more like aleph₀, or more like the continuum, with respect to measure and additivity. For example, we may still want to inquire about our favorite measures, but with uncountable cardinals.

The answer is saturated with set-theoretic independence. For example, it is known to be consistent with the axioms of set theory that Lebesgue measure can be better than countably additive! Under Martin's Axiom (MA), when the Continuum Hypothesis fails, then the union of κ many measure zero sets remains measure zero, for any κ below the continuum. It follows from this that Lebesgue measure is literally ≤κ additive, in the sense that the measure of the union of κ many disjoint sets is the sum of their individual measures (since only countably many of them can have positive measure). This includes the case of uncountable κ, and so goes strictly beyond countable additivity. There are similar results concerning the additivity of the ideal of meager sets.

Indeed, there is a rich subject investigating this called cardinal characteristics of the continuum. I discussed some of the concepts in this MO answer. The point of this subject is to investigate exactly how the dichotomy between countable and continuum plays out in situations when CH fails. Researchers in this area define a number of cardinal invariants, such as:

The bounding number b is the size of the smallest unbounded family of functions from ω to ω. There is no function that bounds every member of the family.

The dominating number d is the size of the smallest dominating family of functions ω to ω. Every function is dominated by a member of the family.

The additivity number for measure is the smallest number of measure zero sets whose union is not measure zero.

The covering number for measure is the smallest number of measure zero sets whose union is all of R.

The uniformity number for measure is the size of the smallest non-measure zero set.

The cofinality number for measure is the smallest size of a family of measure zero sets, such that every measure zero set is contained in one of them.

Each of these numbers is ω₁ under the Continuum Hypothesis, and this expresses the preoccupation with ω that you mention in your question. However, when CH fails, then one cannot prove that any of these numbers is equal to another. Each of them expresses a fundamental characteristic of the continuum, and there are models of set theory distinguishing any two of them (and distinguishing them from ω₁ and the continuum as well).

One can define similar numbers using the ideal of meager sets in place of the ideal of measure zero sets, and the relationships between all these cardinal characteristics are precisely expressed by Cichon's diagram. In particular, no two of them are provably equal, and there are models of set theory exhibiting wide varieties of possible relationships. There are dozens of other cardinal characteristics, whose relationships are the focus of intense study by set theorists working in this area. The main tool for separating these cardinal characteristics is the method of forcing and especially iterated forcing.

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Another answer to your question is to go well beyond the continuum, and perhaps this is what you are really asking about. The analogue of the theory of cardinal characteristics of the continuum has been carried out for arbitrary cardinals κ, and again, the situation is saturated with independence results, proved by forcing.

For certain types of questions, however, it is interesting to note that there is no intermediate possibility between the countable and large cardinals. For example, an ultrafilter on a set can be viewed as an ω-complete 2-valued measure on subsets of ω (ω complete = additive for unions of size less than ω = finitely additive). Can one have an Aleph₁ additive ultrafilter on a set? Well, if F is such an ultrafilter, then it must also be Aleph₂ additive, Aleph₃ additive, and so on, for quite a long way, up to the least measurable cardinal. That is, it is just not possible to have a δ-additive ultrafilter, when δ is uncountable, unless it is also additive up to measurable cardinal. Other similar phenomenon surround the question: does every κ additive filter extend to a κ additive ultrafilter on a set? This is true for κ = ω, since this just amounts to being finitely additive, which is what being a filter means. But for uncountable κ, it is equivalent to the assertion that κ is strongly compact, which is very high in the large cardinal hierarchy.

Answer 2

A related topic: A $\sigma$-smooth linear funtional $L$ on $C(X)$, the set of continuous real-valued functions on a topological space $X$, is a linear functional such that: if $f_n$ is a sequence that decreases pointwise to $0$, then $L(f_n) \to 0$. A generalization is $\tau$-smooth linear functional: if $\mathcal A$ is a family of nonnegative functions in $C(X)$ that is directed in the sense: for any $f_1, f_2 \in \mathcal A$ there exists $f_3 \in \mathcal A$ with $f_3 \le f_1$ and $f_3 \le f_2$, then $\inf_{f \in \mathcal A} L(f) = 0$. For some topological spaces, every $\sigma$-smooth functional is $\tau$-smooth. For other topological spaces, the two concepts are different.