Monday, 5 June 2017

measure theory - A function that is not Lebesgue integrable on [0,1]

I'm currently studying for my measure theory final, and I am struggling with a question:




Give an example of a Borel-measurable function X:(0,1)R that satisfies:



For every a(0,1), the function X is Lebesgue-integrable on [a,1],



The limit lim exists and is finite, with \lambda the standard Lebesgue measure,



The function X is not Lebesgue-integrable on (0,1].



I hope someone could help me out,




thanks.

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