Sentence examples for measurable subsets of from inspiring English sources

Norms in Hilbert spaces are denoted by ∥ ⋅ ∥ H α, ∥ ⋅ ∥ 0 is used to denote the supremum norm of real functions, i.e., for f : R → R we have ∥ f ∥ 0 = sup z ∈ R | f ( z ) |, and | ⋅ | denotes either the absolute value for scalars or the Lebesgue measure for measurable subsets of Euclidean space.

Banach showed in 1923 that Lebesgue measure is not the unique rotation invariant finitely additive probability measure on the measurable subsets of S1.

We also establish the extensions of this inequality to maps with values in measurable subsets of a measure space and to maps with values in subspaces of a linear space.

Specifically, suppose μ a measure on ( T, B ( T ) ) ( B ( T ) is the σ-algebra of Lebesgue measurable subsets of T) which is equivalent to the Lebesgue measure and verifies ∫ T r x ( t, t ) d μ ( t ) < ∞ (9).

Let, let be the Lebesgue measurable subsets of, and let be the Lebesgue measure on.

Accordingly, (mathscr {F}) is taken to be the σ-algebra of Lebesgue measurable subsets of Ω, and μ the Lebesgue measure restricted to (mathscr {F}).

Then there is a constant (C>0) depending only on p and on the weight W such that if (f inmathbb{GT}_{n}) ((1 leq n inmathbb{R}^)) with each (r_{j} geq2) in its representation (1.1) and E is a measurable subset of ([0, 2pi]) of measure at most (lambdain 0, 1]), then int_{[0, 2pi]} f^{p} W leq C^{1+nlambda} int_{[0, 2pi]setminus E} f^{p} W. (4.1).

Let be a Lebesgue measurable subset of and denote the Lebesgue measure of.

For simplicity, we assume that Ω is a bounded Lebesgue measurable subset of ({mathbf{R}}^{n}) with Lebesgue measure equal to 1 and origin lies in Ω.

Let Ω be a measurable subset of a compact group G of positive Haar measure.

This note shows how to solve a mechanism design problem and how to obtain revenue equivalence in a multi-agent quasilinear environment, where each agent's type space is a measurable subset of the real line, not necessarily convex nor finite.