Measurable Set

measurable set

[′mezh·rə·bəl ′set]
(mathematics)
A member of the sigma-algebra of subsets of a measurable space.

Measurable Set

 

(in the original meaning), a set to which the definition of measure given by the French mathematician H. Lebesgue is applicable. Measurable sets are one of the principal concepts of the theory of functions of a real variable and are the most important and an extremely broad class of point sets. In particular, closed sets and open sets lying on some segment are measurable sets. In the abstract theory of measure, sets belonging to the domain of definition μ are said to be measurable with respect to some measure μ. In the case when μ is a probability distribution, measurable sets are also called random events.

References in periodicals archive ?
This needs to change, and while industry must be supported by promoting a growth-oriented environment, a clear and measurable set of targets must be set for sectors that have a realistic potential to be globally competitive, such as information technology and pharmaceutical manufacturing.
for any measurable set E [subset] R with the Lebesgue measure [absolute value of (E)] < [delta].
(ii) F is weakly measurable and there exists a measurable set E [subset] [a, b] with [mu]([a, b] - E) = 0 such that F(E) is separable.
"If you're gonna create a section, be damn sure those products adhere to a measurable set of standards," he says, adding that Patron has a companywide focus on educating trade mixologists about its environmentally conscious ways, which includes a state-of-the-art reverse-osmosis system of water treatment.
Assume that S [??] X x X is a measurable set. Then the family of sections [S.sub.x] = {y [member of] X : (x, y) [member of] X} contains at most continuum of distinct sets and consequently the diagonal D = {(x, x) : x [member of] X} is no longer a measurable set (see [2], p.
Next we prove that [[OMEGA].sub.P] is a measurable set. For the latter, let [psi] : N [right arrow] [F.sub.0] be an enumeration of [F.sub.0].
Since Y is a separable Banach space, to use von Neumann selection theorem, it is enough to show that the graph of H, {(t, y) [member of] [OMEGA] x Y: y [member of] H(t)} = {(t,y) [member of] [OMEGA] x Y: [parallel]f (t) - y[parallel] = d(f (t), Y) and [parallel]y[parallel] [less than or equal to] d(0,[P.sub.Y] (f (t))) + [epsilon]} is a measurable set in the product space.
Let G be a bounded measurable set, then meas(G+(8)) and meas(G"(8)) are continuous functions of 8.
You need a measurable set of variables to observe in order to show improvement over time.
where A [member of] F is an arbitrary measurable set.
However, defining legacy is problematic especially if conceived as an entirely predictable or measurable set of objectives.
if B is any [T.sup.-1][SIGMA] measurable set for which [[integral].sub.B] fd[lambda] converges, we have [[integral].sub.B] fd[lambda] = [[integral].sub.B]E(f)d[lambda].