uniqueness of μ in the Hausdorff moment problem follows from the Weierstrass approximation theorem, which states that polynomials are dense under the uniform norm in the space of continuous functions on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see Carleman's condition, Krein's condition and . There are distributions, such as log-normal distributions, which have finite moments for all the positive integers but where other distributions have the same moments. Variations An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments (for a finite k). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory. See also: Chebyshev–Markov–Stieltjes inequalities and . See also Stieltjes moment problem Hamburger moment problem Hausdorff moment problem Moment (mathematics) Carleman's condition Hankel matrix References (translated from the Russian by N. Kemmer) (Translated from the Russian by D. Louvish) Mathematical analysis Hilbert space Probability problems Moment (mathematics) Mathematical problems Real algebraic geometry Optimization in vector spaces