Measure theory

1. Apr 20, 2014

Funky1981

If f be a measurable function. Assume that

lim λm({x|f(x)>λ}) exists and is finite as λ tends to infinite

Does this imply that ∫|f|dm is finite?

Here m is the Lebesgue measure in R

If not can anyone give me an example??

2. Apr 20, 2014

Erland

f(x)=1/x, for x>0, is a counterexample.

3. Apr 20, 2014

micromass

4. Apr 20, 2014

Funky1981

Thanks for your example. But I cannot convince myself to understand the measure of your case here. Is the measure m({x|1/x>λ}) equal to 0 ??

5. Apr 20, 2014

micromass

No, it's not.

Can you write $\{x~\vert~1/x>\lambda\}$ in a more convenient way that allows you to see easily what the Lebesgue measure is?

6. Apr 20, 2014

Funky1981

i would write it into union of $\{x~\vert~1/x>1/n}$??

7. Apr 20, 2014

micromass

Look at a graph. The set has a really easy structure.

8. Apr 20, 2014

Funky1981

1/x> 0 for all x >0 then so the measure should be infinte. but now why lim λm({x|f(x)>λ}) exists and is finite

9. Apr 22, 2014

Erland

As Micromass wrote: try to rewrite the condition 1/x > λ in a way such that the measure of the corresponding set is easily seen:

If 1/x > λ, what can you then say about x, in terms of λ?