GATE 2021 MA – Question 64

Real Analysis · Measure and Integration · 2 marks · Numerical answer

For x∈(0,1), let n be the number of zeros immediately after the decimal point before the first nonzero decimal digit. Define f(x)=0 for rational x and f(x)=18n for irrational x. Evaluate the Lebesgue integral ∫₀¹f(x)dx.

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Correct answer: 2 to 2

Explanation

Rational points are null and do not affect the integral. Exactly n leading zeros corresponds to an interval of length 9/10^(n+1). Therefore ∫f=Σₙ₌₀∞18n·9/10^(n+1)=(162/10)[(1/10)/(1−1/10)²]=**2**. Endpoint decimal ambiguities also affect only a null set.

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