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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.
It is not difficult to see [29] that the Haar measure coincides with the Lebesgue measure in (mathbb {R}^3).
where (C g,mathbb {P})=1/(nmu (mathbb {P} mathcal {I}(g))) is a normalizing constant and μ denotes the Lebesgue measure in (mathbb {R}^{n}).
Here and in the sequel by ({mathcal H}^k), (k=0,1,ldots,n), we denote the k-dimensional Hausdorff measure in (mathbb {R}^n).
Then there exist (eta>0) and a set A of positive measure in (mathbb {R}^{N}) such that (u(x ge C_{5} Vert u Vert _{L^{p^(mathbb {R}^{N})}+eta) for (xin A).
By (mu^{k}), (kin mathbb {N}), we denote the k-dimensional Lebesgue measure in (mathbb {R}^{k}); by μ we denote the spherical measure on the unit sphere (S^{d-1}) and the measure on the projective space (mathbb{P}^{d-1}).
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For (1le p< infty ) and a vector measure (mu ) in (mathbb {R}), we say that a point a is right j-regular (respectively, left j-regular), if there exist a right completion (overline{mu }) (respectively, left completion) of (mu ) in [a, b] and (j
For each (1le pmeasure (mu ) in (mathbb {R}), consider the Banach space (prod _{j=0}^k L^{p}(Delta,mu _j)) with the norm begin{aligned} Vert fVert _{prod _{j=0}^k L^{p}(Delta,mu _j)} = left( sum _{j=0}^k left| f_jright| _{L^{p}(mu _j)}^p right) ^{1/p} end{aligned}for every (f=(f_0,ldotsdots,f_k)in prod _{j=0}^k L^{p}(Delta,mu _j)).
end{aligned} (2.2 So, E has finite perimeter in (Omega ) if and only if (D{chi _{_{E}}}) is a Radon measure with values in (mathbb {R}^n) and finite total variation.
end{aligned} In addition, for (tauin[gamma, 1)) and (xi_{0} inOmega setminus Omega_{0}), the derivative Xu has the modulus of continuity (r rightarrow r^{tau}+M r)) in a neighborhood of (xi_{0}). It is worth pointing out that the Haar measure in Carnot groups (mathbb {G}) with the underlying manifold (mathbb{R}^{n}) is just the Lebesgue measure in (mathbb{R}^{n}).
Then (varphi ) can be written as a sum of a harmonic function and the potential begin{aligned} U^{mu } z)=int _{mathbb C}log |zeta -z|dmu (zeta ), end{aligned}where (mu ) is a positive measure with compact support in (mathbb C) such that (mu =Delta varphi /2pi ) near (z_0).
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