mérfüggvény

A mérfüggvény, often translated as "measure function" or "measuring function" in English, is a concept in mathematics, particularly in measure theory. It is a function that assigns a non-negative real number, or infinity, to subsets of a given set. This assigned number is intended to represent the "size" or "measure" of the subset in a consistent way.

The fundamental properties of a mérfüggvény are typically: non-negativity, meaning the measure of any set is greater than or equal to zero. It also has the property of empty set measure, where the measure of the empty set is zero. A key characteristic is countable additivity, meaning that for a countable collection of disjoint sets, the measure of their union is equal to the sum of their individual measures. This property is crucial for extending the notion of length, area, or volume to more complex sets.

The most well-known example of a mérfüggvény is the Lebesgue measure on the real line, which assigns to an interval its length. Other examples include counting measures, which assign to a set the number of elements it contains, and probability measures, which assign values between 0 and 1 to events, with the total measure of the sample space being 1. The development of mérfüggvény paved the way for modern integration theory and advanced probability theory.