CanonicalMeasure
plain-language theorem explainer
CanonicalMeasure enumerates the five measures that ground integration and probability in Recognition Science: Lebesgue, counting, Dirac, Hausdorff, and Borel. A researcher formalizing J-cost compatibility with standard measures would cite this enumeration when certifying the configuration dimension equals five. The declaration is a direct inductive definition that derives finite type and decidable equality to support immediate cardinality verification downstream.
Claim. The set of canonical measures consists of the Lebesgue measure, the counting measure, the Dirac measure, the Hausdorff measure, and the Borel measure.
background
In Recognition Science, measure theory supplies the foundation for probability and integration by treating the J-cost function as a measure-theoretic object. The module states that five canonical measures correspond exactly to the configuration dimension D equal to 5, with the Lebesgue measure on the unit interval serving as the canonical normalized measure. J-cost is continuous for positive radii and therefore measurable, and the module records that the Lean development introduces no axioms or sorry statements.
proof idea
This is an inductive definition that lists the five measures explicitly and derives the instances DecidableEq, Repr, BEq, and Fintype automatically via the deriving clause.
why it matters
The definition supplies the enumeration required by the downstream theorem canonicalMeasureCount, which proves that the cardinality is exactly five, and by the structure MeasureTheoryCert, which records both the cardinality and the non-negativity of J-cost for positive radii. It directly implements the module claim that five canonical measures equal configDim D = 5 and thereby closes the foundational step linking J-cost to standard measures in the Recognition Science framework.
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