nLab
real-valued-measurable cardinal

Contents

Idea: the Banach–Ulam problem

The Banach–Ulam problem asks: are there any nontrivial measures on a set XX equipped with the σ-algebra 2 X2^X of all subsets of XX?

All measures are by definition countably additive.

Here a measure on (X,2 X)(X,2^X) is trivial if it can be obtained by the following construction. Start with a set XX, a map f:XRf\colon X\to\mathbf{R}, and a σ-ideal? II on XX. Set μ(A)= xAf(x)\mu(A)=\sum_{x\in A}f(x) if AIA\in I and μ(A)=\mu(A)=\infty otherwise. Then μ\mu is a measure on (X,2 X)(X,2^X).

Once trivial measure on (X,2 X)(X,2^X) are excluded, we can further reduce to the case of probability measures on (X,2 X)(X,2^X) that vanish on all singleton subsets of XX.

Recall that the additivity of a poset PP is the smallest cardinal κ\kappa such that PP has a subset of cardinality κ\kappa without an upper bound.

Recall that the additivity of a measure μ\mu is the smallest cardinal κ\kappa such that there is a disjoint family of cardinality κ\kappa of measurable subsets such that the additivity property of μ\mu fails for this family.

Recall that any measure μ\mu on (X,Σ)(X,\Sigma) induces a σ-ideal? N μN_\mu such that AN μA\in N_\mu if AXA\subset X and there is BΣB\in\Sigma such that ABA\subset B and μ(B)=0\mu(B)=0.

Definitions

A cardinal κ\kappa is real-valued-measurable if there is a κ\kappa-additive probability measure μ:2 κR\mu\colon 2^\kappa\to\mathbf{R} that vanishes on all singleton subsets of κ\kappa.

If the probability measure only takes values 0 and 1, then κ\kappa is known as a measurable cardinal.

A cardinal κ\kappa is atomlessly-measurable if there is an atomless κ\kappa-additive probability measure μ:2 κR\mu\colon 2^\kappa\to\mathbf{R}.

Here an atom of a measure μ\mu on (X,Σ)(X,\Sigma) is AΣA\in\Sigma such that μ(A)0\mu(A)\ne0 and if BΣB\in\Sigma satisfies BAB\subset A, then either μ(B)=0\mu(B)=0 or μ(AB)=0\mu(A\setminus B)=0.

Ulam’s theorem

Theorem

(Ulam?.) Suppse (X,2 X,μ)(X,2^X,\mu) is a nontrivial probability space.

  • The additivity of μ\mu equals the additivity of the σ-ideal? N μN_\mu and is a real-valued measurable cardinal.

  • Any real-valued-measurable cardinal is either atomlessly measurable or measurable.

  • Any atomlessly measurable cardinal is weakly inaccessible? and not greater than 𝔠\mathfrak{c}.

  • Any measurable cardinal is strongly inaccessible?.

  • The Lebesgue measure on R\mathbf{R} extends to 2 R2^{\mathbf{R}} if and only if there is an atomlessly measurable cardinal.

References

Last revised on May 15, 2020 at 17:44:59. See the history of this page for a list of all contributions to it.