# Are there any examples of...

• Mar 13th 2009, 01:32 PM
manjohn12
Are there any examples of...
unbounded functions that are Lebesgue integrable?
• Mar 14th 2009, 03:06 AM
Opalg
Yes, for example $f(x) = 1/\sqrt x$ on (0,1).

[If you want a proof that f is Lebesgue integrable, let $f_n(x) = \begin{cases}n&(0 Note that each f_n is integrable (with integral 2 – (1/n)) and that (f_n) increases to f, and use the monotone convergence theorem.]