GATE 2023 MA – Question 47

Real Analysis · Measure and Integration · 2 marks · Multiple select

Let f_n:ℝ→ℝ equal n²(x−n) on [n,n+1/n²], and zero elsewhere. Write m for Lebesgue measure. Which statements hold as n→∞?

supₓ|f_n(x)|→0
∫ℝ|f_n|→0
m({|f_n|>1/2})→0
m({|f_n|>0})→0

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Correct answer: (B) ∫ℝ|f_n|→0; (C) m({|f_n|>1/2})→0; (D) m({|f_n|>0})→0

Explanation

Each triangular ramp reaches height 1, so its supremum remains 1. Its integral is half base times height=1/(2n²). The region above 1/2 has length 1/(2n²), and its nonzero region has length 1/n². All three tend to zero. **B,C,D**.

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