Sentence examples for be a measurable subset from inspiring English sources

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

Let be a measurable subset of.

Let E be a measurable subset of ℝ. Suppose that 1 < p − ( E ) ≤ p + ( E ) < ∞.

We recall also that u and (u_{ast}) are equi-measurable, i.e. mu_{u}(t)=mu_{u_{ast}}(t),quad tinmathbb{R}, which implies that for any non-negative Borel function ψ we have int_{Omega}psibigl u(x bigr), mathrm{d}x=int _{0}^{|Omega|}psibigl u_{ast }(s bigr), mathrm{d}s, and if (EsubsetOmega) be a measurable subset, then int_{E}u(x), dxleqint_{0}^{|E|}u_{ast}(s), ds.

Let (mathcal {M}) be a measurable subset of the star sphere S. We call a random vector (U_{mathcal {M}}) star-generalized uniformly distributed on (mathcal {M}) (or the ({mathcal {M}} -restriction of U) if (U_{M}} -restrictionllofs the distribUtifn law begin{array}rcl@ P^{U_{mathcal{M}}}(A)=follows{S}(A)}{O_{S}(matheal{M})},; Ain mathcal{B}_{S} cap mathcal{M}.

Throughout the paper, we write A ≲ B if there exists a positive constant C, independent of appropriate quantities such as functions, satisfying A ≤ C B. If p ∈ [ 1, ∞ ], the conjugate number p ′ is defined by p ′ = p p − 1 and if p ∈ ( 0, 1 ), the conjugate number p ′ is defined by p ′ = p 1 − p. Let E be a measurable subset of R n and | E | = ∫ E d x.

The symbol Ω will denote a bounded open subset of R n and we shall always consider the case n ≥ 2. With O ⊂ R n being a measurable subset with positive measure, and with g : O → R n, n ≥ 1, being a measurable map, we shall denote by Open image in new window its integral average; here | O | denotes the Lebesgue measure of O.

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).

where is a ball in and is a measurable subset of.

Suppose that E is a measurable subset of ([0,T]) and (G t,x)) is continuous in x for a.e.

It is known that (see, e.g., [32, 33]) if is a measurable function on and is a measurable subset of, then the following inequalities hold: (2.8).