Write χ E for the indicator of E. Let f(x)= n=1(2n/n3) [2-(n+1),2-n](x) on [0,1]. P:f∈Lᵖ[0,1] for every 1<p<2. Q:f∈L¹[0,1]. Select all correct statements.

GATE 2023 MA – Question 48

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

Write χ_E for the indicator of E. Let $f(x)=\sum_{n=1}^\infty(2^n/n^3)\chi_{[2^{-(n+1)},2^{-n}]}(x)$ on [0,1]. P:f∈Lᵖ[0,1] for every 1<p<2. Q:f∈L¹[0,1]. Select all correct statements.

P true
Q true
Q false
P false

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Correct answer: (B) Q true; (D) P false

Explanation

Endpoint overlaps have measure zero. Thus $\int|f|^p=\frac12\sum_{n\ge1}2^{n(p-1)}/n^{3p}$. For p=1 this is (1/2)Σ1/n³<∞. For every p>1 the exponential growth outweighs the polynomial denominator, so terms do not tend to zero and the integral diverges. **B,D**.

GATE 2023 MA – Question 48

Explanation

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