Lebesgue measure


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Lebesgue measure

[lə′beg ‚mezh·ər]
(mathematics)
A measure defined on subsets of euclidean space which expresses how one may approximate a set by coverings consisting of intervals.
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They cover Thurston maps, LattAaAeAeAes map quasiconformal and rough geometry, cell decompositions, expansion, Thurston maps with two or three postcritical points, visual metrics, symbolic dynamics, tile graphs, isotopies, subdivisions, quotients of Thurston maps, combinatorially expanding Thurston maps, invariant curves, the combinatorial expansion factor, the geometry of the visual sphere, relational Thurston maps and Lebesgue measure, a combinatorial characterization of LattAaAeAeAes maps, and outlook and open problems.
An analogous proof also works if we study the functions that are continuous except on a discrete set, a subset of a closed set with finite Lebesgue measure or a bounded set.
assumed absolutely continuous with respect to a Lebesgue measure, that describes the random behavior of the sequence.
To be more precise, let [lambda] x P denote the product of Lebesgue measure and the probability measure on the Cartesian product I x [OMEGA], and let P denote the sigma algebra of [G.
for all [GAMMA] [subset or equal to] [0, 2[lambda]] with m([GAMMA]) [less than or equal to] [delta]([lambda]), where m([GAMMA]) is the Lebesgue measure of [GAMMA].
We reconsider the problem for area given by the Lebesgue measure of the set (which includes the area considered in [1]) and for any given point in the plane.
Note that d(x, y) = [lambda](([phi](x) [union] [phi](y))\([phi](x) [intersection] [phi](y)), where [lambda] is Lebesgue measure on R.
Remark 2: From Propositions 2 and 3, if [beta]([mu]) = [infinity] whenever [mu] is not equivalent to the Lebesgue measure on (0,1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
2]-integrable with respect to the Lebesgue measure on [0, 2[pi]).
Appendices review partially ordered sets, Lebesgue measure theory, and mollifications.
2], and let [omega] be an open polygonal convex subset of [OMEGA] and I [subset] [partial derivative][omega], with [absolute value of I] > 0; [absolute value of I] is the one-dimensional Lebesgue measure of I.