In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.
In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.
==Definition== Let R be a ring of subsets (closed under union and relative complement) of a fixed set X and let \mu_0 : R \to [0, \infty] be a set function. \mu_0 is called a pre-measure if \mu_0(\varnothing) = 0 and, for every countable (or finite) sequence A_1, A_2, \ldots \in R of pairwise disjoint sets whose union lies in R, \mu_0 \left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu_0(A_n). The second property is called \sigma-additivity.
Discovered by embedding cosine similarity (sentence-transformers MiniLM, 384-dim).