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

The hypothesis buys exactly two powers

Let $μ$ denote the Lebesgue measure on $R$ and $f : R \to [0, \infty)$ be a Lebesgue measurable function. For n ≥ 1, let $E_{n} =$ { $x \in R : n - 1 \leq f (x) < n$ }. Suppose that $\sum (n \geq 1) n^{2} μ (E_{n}) < \infty$ . Which of the following statements are true?

A. $f \in L^{1} (R)$ ✓
B. $f \in L^{2} (R)$ ✓
C. $f \in L^{4} (R)$
D. $f \in L^{\infty} (R)$

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.

See pricing

50 are analysed free — try those first.

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

The hypothesis buys exactly two powers

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