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Counting measure is an outer measure for which every subset is measurable

Example

For every set $X$ , counting measure is an outer measure on $P (X)$ and every subset of $X$ is Carathéodory measurable.

Facts & Assumptions

Given: A set $X$ and its counting set function $#_{X}$ .

[L1]

For every set $X$ , counting measure is a measure on $(X, P (X))$ . (Counting measure is a measure)

[F1]

A set $E \subseteq X$ is Carathéodory measurable for $μ^{*}$ when $μ^{*} (A) = μ^{*} (A \cap E) + μ^{*} (A ∖ E)$ for every $A \subseteq X$ . (Carathéodory measurable sets)

Verification

technique · direct

1.1

The measure axioms in [L1] give normalization and countable additivity on the full power set, hence monotonicity and countable subadditivity; thus counting measure is an outer measure.

2.1

For arbitrary $A, E \subseteq X$ , the decomposition $A = (A \cap E) ⊔ (A ∖ E)$ and countable additivity in [L1] give $#_{X} (A) = #_{X} (A \cap E) + #_{X} (A ∖ E)$ , so [F1] holds for every $E$ , including finite, infinite, and empty cases.

L1F1algebra∎

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. Folland, Real Analysis, 2nd ed., Section 1.4 (standard reference, not scraped)