Measurable Chromatic Number of the Plane

As you know, the length of a segment [a, b], a < b, on the line E1 is defined as b − a. The area A of a rectangle [a1, b1] × [a2, b2], ai < bi, in the plane E2 is defined as A = (b1 − a1)(b2 − a2). The French mathematician Henri Léon Lebesgue (1875–1941) generalized the notion of area to a vast class of plane sets. In place of area, he used the term measure. For a set S in the plane, we define its measure μ∗(S) as follows:with the infimum taken over all coverings of S by a countable sequence {Ri} of rectangles. When the infimum exists, S is said to be Lebesgue-measurable or – since we consider here no other measures – a measurable set – if for any set B on the plane μ∗(B) = μ∗(B ∩ S) + μ∗(B\S). For a measurable set S, its measure is defined by μ(S) = μ ∗ (S).