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Prove that

If $f$ is a monotone function which defined on interval, then $f$ is measurable.

Proof: If $f$ is increasing and define on interval, then the set $A=\left\{x:f(x)>a\right\}$ will be an interval for all a, and it's measurable since each interval is measurable.

Does this an accepted proof? Need I to show that A is an interval? If yes; how can I prove that?

Souza
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Duaa Hamzeh
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  • The defining property of an interval is that if $a,b$ are in the interval, so is any point between them. Can you show that $A$ has this property? In fact, $A$ has a stronger property : if a point is in $A$, so are all the points to the right of it in the domain. Since the domain itself is an interval,you can conclude. – Sarvesh Ravichandran Iyer Oct 26 '18 at 05:28

1 Answers1

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Suppose $f$ is increasing. $A$ is an interval; because $x\in A$ and $y >x$ implies $f(y) \geq f(x) >a$ so $y \in A$. This means $A$ contains all points from its infimum to $\infty$, so $A$ is an interval. A similar argument works if $f$ is decreasing.

Kavi Rama Murthy
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