Sentence examples for be a measurable set from inspiring English sources

Let (Esubsetmathbb{R}^{N}) be a measurable set of finite measure and let (z_{j} Erightarrowmathbb{R}^{N}) be a sequence of measurable functions.

Let ν be a positive measure and Ω be a measurable set with (nu(Omega)=1).

Let E be a measurable set in ℝ with positive measure.

Let ν be a positive measure and let Ω be a measurable set with (nu(Omega)=1).

Let be a measurable set such that and in.

Let (Esubset mathbb {H}) be a measurable set.

Definition 5 Let ( Ω, A ) be a measurable space and F : Ω ⇉ R k be a measurable set-valued map.

If μ is a measure on (mathbb{R}), and if (mathcal{S}subseteq mathbb{R}) is a measurable set of positive measure, then the integral mean point c=frac{1}{mu(mathcal{S})}int_{mathcal{S}}x,dmu (4) is called the barycenter of the set (mathcal{S}) respecting measure μ, or just the set barycenter.

It is easy to see that E k = x k + μ(E k )A k, where A k is a measurable set with measure 1. Henceforward, we assume the sets A k are uniformly bounded.

If μ is a finite measure on I and A ⊆ I is a measurable set with the positive measure, then we define the integral φ-quasi-arithmetic mean on the set A with respect to the measure μ by M φ ( A, μ ) = φ − 1 ( 1 μ ( A ) ∫ A φ ( t ) d μ ( t ) ). (4.4).

Equivalently, if E is a measurable set of finite measure in (mathbb {R}^n) then begin{aligned} |E|le omega _nleft( frac{mathrm{diam,}(E)}{2}right) ^n, end{aligned}with the equality holding if and only if E is a ball.