We consider compact sets A ⊂ ⊂ R p with the Lebesgue measure λ λ ( A ) > 0 and s ⩾ ⩾ p . For every N ⩾ ⩾ 2 fix an N -point s -equilibrium configuration ω ω N ∗ ∗ = { x 1 , N , … … , x N , N } . Set μ μ N := 1 N ∑ ∑ i = 1 , … … , N δ δ x i , N , where δ δ x is a unit point mass at point x . Under these assumptions, in the sense of weak convergence of measures, μ μ N → → ∗ ∗ μ μ , where μ μ is the Lebesgue measure restricted to A ; i.e., μ μ ( B ) = λ λ ( A ∩ ∩ B ) / λ λ ( A ) . Furthermore, it is true that lim N → → ∞ ∞ E s ( A , N ) N 1 + s / p = C s , p λ λ ( A ) s / p , where the constant C s , p does not depend on the set A and, therefore, C s , p = lim N → → ∞ ∞ E s ( [ 0 , 1 ] p , N ) N 1 + s / p , where [ 0 , 1 ] p is the unit cube in R p .