Mittetäielikkus

Mittetäielikkus is a term seldom used in Finnish mathematics. It is a compound formed from mitta (measure) and täielikkus (completeness) and may appear in informal discussions or some educational texts to denote a relationship between measurability and a completeness property. Because the term is not widely established, its exact meaning depends on the context in which it is used.

In standard measure theory, the precise concept is called the completeness of a measure space. A measure space (X, Σ, μ) is complete if every subset of every μ-null set is measurable, i.e., belongs to Σ. This property ensures there are no non-measurable subsets contained entirely within sets of measure zero. Completeness is a key condition for handling null sets and for defining the completed σ-algebra associated with a given measure.

Examples help illustrate the idea. The Lebesgue measure on the real line is complete: all subsets of Lebesgue-null sets are Lebesgue-measurable. By contrast, starting from the Borel σ-algebra with the Lebesgue measure, the space is not complete, but its completion (the Lebesgue σ-algebra) is complete.

Terminology-wise, mittetäielikkus may be used informally to describe this notion, but the standard Finnish terminology for the concept is typically “mitta-tila on täydellinen” or “completeness of a measure space.” When encountered, it is advisable to define the term explicitly in the given text and refer to the well-established concept of measure completeness. See also measure, σ-algebra, null set, Lebesgue measure, and completion of a measure.