Measurable set

A subset of a measurable space $(X,\mathcal A)$ belonging to $\mathcal A$, where $\mathcal A$ is a ring or $\sigma$-ring of subsets of $X$. The concept arose and was developed in the process of the solution and generalization of the measurement of areas (lengths, volumes) of various sets; that is, the problem of the extension of area (length, volume) as an additive function of polygons (segments, polyhedra) to a wider system of sets. A measurable set was defined to be a set in the system to which the extension can be realized; this extension is said to be the measure. Thus were defined the Jordan measure, the Borel measure and the Lebesgue measure, with sets measurable according to Jordan, Borel and Lebesgue, respectively. The solution of the problem of extending any fixed measure in $\mathbf R^n$ led to the Radon measure (Lebesgue–Stieltjes measure) and sets measurable with respect to the Radon (Lebesgue–Stieltjes) measure. The measurable sets connected with a measure defined on an abstract set are the sets on which the measure under discussion is defined.

References

[1]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802

References

[a1]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202