This book intends to be a basic approach to measure theory.
In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the base set. Depending on the application, the "size" of a subset may be interpreted as (for example) its physical size, the amount of something that lies within the subset, or the probability that some random process will yield a result within the subset. The main use of measures is to define general concepts of integration over domains with more complex structure than intervals of the real line. Such integrals are used extensively in probability theory, and in much of mathematical analysis.
It is often not possible or desirable to assign a size to all subsets of the base set, so a measure does not have to do so. There are certain consistency conditions that govern which combinations of subsets it is allowed for a measure to assign sizes to; these conditions are encapsulated in the auxiliary concept of a σ-algebra.
Measure theory is that branch of real analysis which investigates σ-algebras, measures, measurable functions and integrals.
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- Tao, Terence: An introduction to measure theory
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