Integration and Lipschitz functions

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 ²ⁿ _k=1 |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 ) ^k=1,...,2n _n≧0 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).