Sentence examples for of finite measure 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.

Under this assumption there is an increasing sequence of measurable sets of finite measure whose union equals Ω.

Conversely, provided that (sup_{n in mathbb {N}} int vert f_{n}vert,dmu< infty), equi-integrability implies uniform integrability on each measurable set of finite measure; see [6], Proposition 2.8, for related results.

Using some previous results proved in collaboration with Hayman and Weitsman [84], he showed that there exists a constant C(n) such that for all measurable sets of finite measure begin{aligned} alpha (E ^4 le C(n) D(E).

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.

There exists a constant (gamma (n)) such that for any measurable set E of finite measure begin{aligned} alpha (E ^2 le gamma (n)D(E).

Let denote the δ-ring of subsets of finite measures.

We further prove the concentration of (mathcal{S}_{h}^{W}(f)) in arbitrary sets of finite measures.

The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures.

Let X be the set of finite measures on S. For μ, ν ∈ X, write μ ⪯ ν if μ ( B ) ≤ ν ( B ) for each B ∈ S. Then ⪯ is a partial order.

Our first example shows that Theorem 3.1 is a strict generalization of Corollaries 4.2 and 4.3 even in the case of a finite measure.