GATE 2026 MA – Question 13
Let C be the Cantor set. On [0,1], define f(x) = x²⁰²⁶ on C; cos(πx) on [0,1/2]∖C; sin(πx) on [1/2,1]∖C. Find its Lebesgue integral.
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Correct answer: (A) 2/π
Explanation
C has Lebesgue measure zero, so its assigned values do not affect the integral. Thus $\int f=\int_0^{1/2}\cos\pi x\,dx+\int_{1/2}^1\sin\pi x\,dx=1/\pi+1/\pi=\mathbf{2/\pi}$, **A**.