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Given a function $f(x)$ over any closed interval $[a,b]$, does there is any if and only if condition so that function will be Lebesgue Integrable.

For Riemann integrability there is a condition that

A function is Riemann integrable on $[a,b]$ iff the function is continuous almost everywhere.

Thankyou

Angina Seng
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gaurav saini
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  • On a bounded interval or more generally a set of finite measure a bounded function is Lebesgue integrable if and only if it is measurable. – RRL May 12 '19 at 06:03
  • it doen't matter whether the set is open or closed? – gaurav saini May 12 '19 at 06:10
  • Other statements can be made for unbounded functions and/or infinite measure sets which are not as concise and do not have a Riemann counterpart. The Riemann integral is not defined in these cases. – RRL May 12 '19 at 06:11
  • The set over which the integral is taken can be open or closed or neither. It has to be measurable though. This may be of interest. – RRL May 12 '19 at 06:13
  • Thankyou for your answer. – gaurav saini May 12 '19 at 06:16
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    A good reference is Real Analysis by Royden. – RRL May 12 '19 at 06:19

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