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Integration and Lipschitz functions

Authors:	Piotr Niemiec

Institution:	(1) Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Abstract:	The aim of the paper is to prove that every f ∈ L ¹(0,1]) is of the form f = $\sum\nolimits_{n = 0}^\infty {\sum\nolimits_{k = 1}^{2^n } {a_{n,k} \frac{{j_{n,k} }} {{\left\\| {j_{n,k} } \right\\|}}} }$ , where j _n,k is the characteristic function of the interval k- 1 / 2ⁿ, k / 2ⁿ) and Σ_n=0^∞Σ_k=1²ⁿ\|a _n,k\| is arbitrarily close to \|\|f\|\| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on 0,1], then for any ɛ > 0 there exists a sequence (b _n,k)_n≧0^k=1,...,2n of real numbers such that $\sum\nolimits_{n = 0}^\infty {\sum\nolimits_{k = 1}^{2^n } {\left\| {b_{n,k} } \right\| \leqslant \int_0^1 {td\mu (t) + \varepsilon } } }$ and $\int_0^1 {g(t)d\mu (t) = g(0) + \sum\nolimits_{n = 0}^\infty {\sum\nolimits_{k = 1}^{2^n } {b_{n,k} 2^n (g(\tfrac{k} {{2^n }}) - g(\tfrac{{k - 1}} {{2^n }}))} } }$ for each Lipschitz function g: 0,1] → ℝ (Theorem 3).

Keywords:	integration integrable functions Lipschitz functions
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