logmeasure

Logmeasure is a functional concept derived from a base measure μ defined on a measurable space (X, Σ). For any measurable set A ∈ Σ, the log-measure is defined by logmeasure(A) = log μ(A) when μ(A) > 0; if μ(A) = 0, logmeasure(A) is set to −∞ in the extended real sense. It is not a measure: for disjoint A and B, logmeasure(A ∪ B) = log(μ(A) + μ(B)) generally differs from log μ(A) + log μ(B). Consequently, logmeasure is not additive and does not satisfy σ-additivity. The log-measure is monotone with respect to A because μ is monotone.

It is primarily used as a transformation of a measure rather than as a stand-alone measure. The log operation converts multiplicative structures into additive ones: for positive scalars c, log(c μ(A)) = log c + log μ(A). In statistical contexts, the term log-measure is sometimes used informally to refer to the logarithm of a probability measure or density, which gives rise to log-likelihood functions and log-densities. These are central in likelihood-based inference, where products of probabilities across independent observations become sums of their logarithms.

Examples: If μ is the counting measure on a finite set X, then for A ⊆ X with |A| = k > 0, log-measure(A) = log k. If μ is the Lebesgue measure on R and A has finite positive length, log-measure(A) = log length(A).

See also: log-likelihood, log-density, logarithmic transformation.