The modern notion of measure, developed in the late 19th century, is an extension of the notions of length, area or volume. A measure m is a law which assigns a number $m(A)$ to certain subsets $A$ of a given space and is a natural generalization of the following notions: 1) length of an interval, 2) area of a plane figure, 3) volume of a solid, 4) amount of mass contained in a region, 5) probability that an event from $A$ occurs, etc.

It originated in the real analysis and is used now in many areas of mathematics like, for instance, geometry, probability theory, dynamical systems, functional analysis, etc.

Reference: Measure Theory

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