Locally finite measure¶
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.
Core Idea¶
Locally finite measure is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. A measure/signed measure/complex measure \mu defined on \Sigma is called locally finite if, for every point p of the space X, there is an open neighbourhood Np of.
Scope of Application¶
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Definition. A measure/signed measure/complex measure \mu defined on \Sigma is called locally finite if, for every point p of the space X, there is an open neighbourhood Np of p.
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Definition. In more condensed notation, \mu is locally finite if and only if.
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Definition. \text{for all } p \in X, \text{ there exists } Np \in T \mbox{ such that } p \in Np \mbox{ and } \left|\mu\left(Np\right)\right|.
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Examples. Any probability measure on X is locally finite, since it assigns unit measure to the whole space.
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Examples. Similarly, any measure that assigns finite measure to the whole space is locally finite.
Clarity¶
A clear use of Locally finite measure names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.
Manages Complexity¶
Locally finite measure compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—in more condensed notation, \mu is locally finite if and only if.—and the practical consequence—the counting measure is sometimes locally finite and sometimes not: the counting measure on the integers with their usual discrete topology is locally finite, but the counting measure on the real line.
Abstract Reasoning¶
- Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.
- Check operation and conditions. \text{for all } p \in X, \text{ there exists } Np \in T \mbox{ such that } p \in Np \mbox{ and } \left|\mu\left(Np\right)\right|.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Locally finite measure transfers literally when a new case preserves the same carrier type, relation, and recognition test. A measure/signed measure/complex measure \mu defined on \Sigma is called locally finite if, for every point p of the space X, there is an open neighbourhood Np of p such that the \mu -measure of Np is finite. In more condensed notation, \mu is locally finite if and only if. Beyond the home domain. No canonical parent is asserted for Locally finite measure.
Relationships to Other Abstractions¶
Current abstraction Locally finite measure Domain-specific
Parents (1) — more general patterns this builds on
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Locally finite measure is a kind of Measure Prime
A locally finite measure is a measure assigning finite measure to a neighborhood of every point.
Hierarchy paths (2) — routes to 2 parentless roots
- Locally finite measure → Measure → Aggregation → Micro Macro Linkage
- Locally finite measure → Measure → Set and Membership
Neighborhood in Abstraction Space¶
Locally finite measure sits in a moderately populated region (43rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Measure-Theoretic Constructions (10 abstractions)
Nearest neighbors
- Counting measure — 0.89
- Borel regular measure — 0.88
- Uniformly distributed measure — 0.88
- Topological Algebra — 0.87
- Filling radius — 0.86
Computed from structural-signature embeddings · 2026-10-08