Let f(x)= n=02-n (n,n+1](x) on R, and define S =\x:f(x)>\. Which is true?

$S_{\sqrt5/2}$ is not measurable.

GATE 2020 MA – Question 41

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

Let $f(x)=\sum_{n=0}^\infty2^{-n}\chi_{(n,n+1]}(x)$ on R, and define $S_\alpha=\{x:f(x)>\alpha\}$. Which is true?

$S_{1/2}$ is open.
$S_{\sqrt5/2}$ is not measurable.
$S_0$ is closed.
$S_{1/\sqrt\pi}$ is measurable.

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Correct answer: (D) $S_{1/\sqrt\pi}$ is measurable.

Explanation

f is a countable sum of measurable indicator functions and is therefore measurable, making every strict superlevel set measurable. In particular $S_{1/\sqrt\pi}$ is measurable, D. Also $S_{1/2}=(0,1]$ is not open, $S_0=(0,\infty)$ is not closed, and $S_{\sqrt5/2}$ is empty and hence measurable.

GATE 2020 MA – Question 41

Explanation

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