Mathematics · Measure Theory
Let μ denote the Lebesgue measure on ℝ. Suppose that f is a non-negative Lebesgue measurable function on ℝ. Let 0 = a0 < a1 < a2 < … be an unbounded sequence such that an+1 ≤ c an for some real number c and for all n ≤ 1. Let Ak = {x ∞ ℝ | ak ≤ f(x) < ak+1} for each k ≤ 0. Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
Share:
- 1f is Lebesgue integrable if and only if f is a constant function.
- 2If ∑k≥0 ak μ(Ak) is infinite, then f can still be Lebesgue integrable on ℝ.
- 3If ∑k≥0 ak μ(Ak) is finite, then f is Lebesgue integrable on &reals].✓
- 4If f is Lebesgue integrable on ℝ, then ∑k≥0 ak μ(Ak) must be infinite.
Correct answer
If ∑k≥0 ak μ(Ak) is finite, then f is Lebesgue integrable on &reals].
Download Single Question PDF
Save this question and verified answer as a clean study sheet with live clickable link.
👥
Share with Friends & Study Groups
Challenge your peer aspirants on WhatsApp or Telegram to solve this question!
Online Test Series
Start Free Mock Test →Preparing for Mathematics?
Practice full-length mock tests with real exam timers, instant score analysis, and all-India ranking.
Chapter Practice
Practice Measure Theory MCQs →Want more practice questions on Measure Theory?
Solve complete topic-wise tests with detailed solutions on GkGet.
Related Practice Questions
Recommended practice questions in Mathematics · Measure Theory