Recent content by Crossfader

  1. C

    Proving that a measurable function is integrable

    1. Homework Statement [/b] Let f:ℝ\rightarrowℝ be measureable and A_{k}=\left\{x\inℝ:2^{k-1}<\left|f(x)\right|≤2^{k}\right\}, k\in \mathbb{Z}. Show that f is integrable only if \sum^{∞}_{k=-∞}2^{k}m(A_{k}) < ∞ . Homework Equations By the definition f is integrable in ℝ if and only if...
  2. C

    Convergent series with non-negative terms, a counter-example with negative terms

    Homework Statement The terms of convergent series \sum_{n=1}^\inftya_n are non-negative. Let m_n = max{a_n, a_{n+1}}, n = 1,2,... Prove that \sum_{n=1}^\inftym_n converges. Show with a counter-example that the claim above doesn't necessarily hold if the assumption a_n\geq0 for all...