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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:

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 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. There are similar results concerning the additivity of the ideal of meager sets.

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.

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.

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.