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Let f be a nonnegative integrable function on measureable space (X,v). Then tv ({x: f (x)>t}) converges to 0, as t goes infinity.

I want to prove this statement. I got that v ({x: f (x)>t}) goes to zero as t goes infinity. But I cannot prove the full statement.

If tv ({x: f (x)>t}) goes to zero, g (t)v ({x: f (x)>t}) converges to zero whenever g (t)t}) converges to zero or not for g (t)>t. I want to know significant difference of g (t) from t, not the form the constant times t.

Please anyone help me.

user277793
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  • You probably mean $g( t)/t$, not $g( t)t$. Finally, the significance of $t$ is that $t \cdot 1_{f(x)>t} \leq |f|$. I leave it to you to find out why this is helpful. – PhoemueX Nov 22 '15 at 09:28

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