Counting measure on an arbitrary set

Depends on

• Finite, countably infinite, countable, uncountable
• The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined

Used by

• A non-sigma-finite premeasure has distinct Borel extensions Counterexample
• A norm need not satisfy the parallelogram law Counterexample
• A pointwise limit of integrable functions need not be integrable Counterexample
• An inner-product space need not be complete Counterexample
• Counting measure on [0,1] shows the dominating measure needs sigma-finiteness Counterexample
• Counting-measure tails decrease to the empty set while every term has infinite measure Counterexample
• Jensen's inequality can fail on an infinite measure space without normalization Counterexample
• Lebesgue plus counting measure has no Lebesgue decomposition relative to Lebesgue measure Counterexample
• Assuming countable choice, counting measure is sigma-finite exactly on countable sets Example
• Counting measure is an outer measure for which every subset is measurable Example
• Counting measure represents finite-support summation on a discrete LCH space Example
• Diagonal operator on ell p is compact iff diagonal tends to zero Example
• Finite counting measure on n points recovers ℝⁿ p-norms Example
• Finite-rank truncations of a square-integrable kernel Example
• Integral operator trace under a valid diagonal hypothesis Example
• Integrating against counting measure recovers a series Example
• Spectrum of the unilateral shift Example
• The standard inner products make K n, ell two and quotient L two Hilbert spaces Example
• Every translation-invariant measure on the Borel sets of ℝ is a nonnegative multiple of Lebesgue measure False statement
• FALSE: continuity from above needs no finiteness hypothesis False statement
• FALSE: every Borel measure on ℝ is finite on compact sets False statement
• FALSE: Jensen's inequality holds on an infinite measure space without normalization False statement
• FALSE: Lᵖ(μ) is separable for every measure μ and every 1 ≤ p < ∞ False statement
• FALSE: pointwise limits of integrable functions are integrable False statement
• FALSE: the extension of a premeasure is always unique False statement
• FALSE: the Radon-Nikodym theorem holds without sigma-finiteness False statement
• Counting measure on a discrete group is Haar, Haar measures there are its multiples, and integrals against them are sums Lemma
• Counting measure is a measure Proposition
• The p-functional need not be a norm for 0 < p < 1 Proposition
• Zero-dimensional Hausdorff measure is counting measure Proposition
• ℓᵖ is the Lᵖ space of counting measure Remark
• Assuming countable choice, every measure is the sum of its semifinite part and a zero-infinity-valued measure Theorem
• ℓᵖ includes into ℓʳ for p < r Theorem

Dependency tree · two levels

13 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

• S. Axler, Measure, Integration & Real Analysis, Example 2.55 (standard reference, not scraped)