the a0 equals 0 block carries no weight so it can hide infinite mass — CSIR NET June 2025 Part C, solved

The a0 equals 0 block carries no weight so it can hide infinite mass

Let $μ$ denote the Lebesgue measure on $R$ . Suppose that f is a non-negative Lebesgue measurable function on $R$ . Let $0 = a_{0} < a_{1} < a_{2} < \dots$ be an unbounded sequence such that $a_{n + 1}$ ≤ ca $_{n}$ for some real number c and for all n ≥ 1. Let A_k = { $x \in R$ | a_k ≤ f(x) < $a_{k + 1}$ } for each k ≥ 0. Which of the following statements are true?

A.If f is Lebesgue integrable on $R$ , then $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite.✓
B.If $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite, then f is Lebesgue integrable on $R$ .
C.If $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite, and $f (x) \geq a_{1}$ for all $x \in R$ , then f is Lebesgue integrable on $R$ .✓
D.If $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite and f is bounded, then f is Lebesgue integrable on $R$ .

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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this