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...
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...