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ABSTRACT:
The current paper presents the concept of measures in mathematics. The terms "measure," "measurable," etc. have very precise technical definitions that can make them appear difficult to understand. Measures are important not only because of their intrinsic geometrical and probabilistic significance, but because they allow us to define integrals. In this paper, we discuss brief theory of measures and their properties. The paper also includes the various important list of examples of the measure that are used in various fields of mathematics. We have used standard definitions and notations from set theory.
KEYWORDS: measure, set, Lebsegue
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INTRODUCTION:
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral. A measure is a mathematical object that quantifies the size of sets. A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. Mathematically, a measure is a function that assigns a non-negative real number to (certain) subsets of a set X. It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets is the sum of the measures of the "smaller" subsets.
At first thought, it comes into mind that there is only one natural measure of a set's size - |its cardinality. Indeed, the cardinality is a measure written as #A and it is a nonnegative natural number. But if we analysis the concept carefully, we feel that there are other significant ways to measure the size of a set [1-3]. Consider the two sets[10,20] and [10,100]. Both have the same cardinality, but our mind suggests that the latter is bigger" somehow. Indeed, we can compute the length of the two intervals, 10 and 90 respectively, which confirms our intuition. So, the length of a set provides a different measure of its size.
2.Definition of Measure:
Let Xbe a set and E a a-algebra over X. A function m from E to the extended real number line is called...