GATE 2023 MA – Question 17
Real Analysis · Measure and Integration · 1 mark · Multiple choice
Measurable E_k⊆[0,1] have measure 1/k². Define $E=\bigcap_n\bigcup_{k\ge n}E_k$ and $F=\bigcup_n\bigcap_{k\ge n}E_k$. P:m(E)=0; Q:m(F)=0. Choose truth values.
- Both true
- Both false
- P true, Q false
- Q true, P false
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (A) Both true
Explanation
The union bound gives $m(E)\le\sum_{k\ge n}1/k^2$ for every n. Its tail tends to zero, so m(E)=0. Since liminf E_k⊆limsup E_k, F⊆E and m(F)=0. **A**.
More Measure and Integration questions
- GATE 2023 MA Q47 – Let f n:ℝ→ℝ equal n²(x−n) on [n,n+1/n²], and zero elsewhere. Write m for Lebesgue measure.…
- GATE 2023 MA Q48 – Write χ E for the indicator of E. Let f(x)= n=1(2n/n3) [2-(n+1),2-n](x) on [0,1].…
- GATE 2022 MA Q38 – Continuous f:[−π,π]→ℝ satisfies f(x)>f(0)/2 for x <δ,0<δ<π. Put P (n,δ)(x)=(1+cos…
- GATE 2021 MA Q64 – For x∈(0,1), let n be the number of zeros immediately after the decimal point before the…
- GATE 2026 MA Q13 – Let C be the Cantor set. On [0,1], define f(x) = x²⁰²⁶ on C; cos(πx) on [0,1/2]∖C; sin(πx)…
- GATE 2024 MA Q14 – Evaluate the source limit 0-1 0 e-x/[3x+x2+x+4]dx.
All Measure and Integration questions →