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Part CCSIR NET December 2024the-hypothesis-buys-exactly-two-powers

The hypothesis buys exactly two powers

Let μ\mu denote the Lebesgue measure on R\mathbb{R} and f:R→[0,∞)f : \mathbb{R} \to [0, \infty) be a Lebesgue measurable function. For n ≥ 1, let En=E_{n} = {x∈R:n−1≤f(x)<nx \in \mathbb{R} : n - 1 \le f(x) < n}. Suppose that ∑(n≥1)n2μ(En)<∞\sum(n\ge1) n^{2}\mu(E_{n}) < \infty. Which of the following statements are true?

  1. A.f∈L1(R)f \in L^{1}(\mathbb{R})✓ Correct
  2. B.f∈L2(R)f \in L^{2}(\mathbb{R})✓ Correct
  3. C.f∈L4(R)f \in L^{4}(\mathbb{R})
  4. D.f∈L∞(R)f \in L^\infty(\mathbb{R})

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests moments and tails.

See pricing

50 are analysed free — try those first.

The trap it tests

Moments and tails

A moment assumed to exist, or tail behaviour assumed to be tame.

Drill statements like this

Related counterexample: L¹[0,1] ⊆ L²[0,1]

More on this topic

The chapter behind this: L^p spaces: inequalities and inclusions — free to read

From Lebesgue Measure and Integration › L^p spaces essentials

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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