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I have a question about normed space and integrable.

If $f$ is in $L_\infty$, $g$, which is $g \le f$, can be absolutely integrable ($g$ is in $L_1$)?

And how can I prove it?

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In view of the two questions you asked on the site so far, I suggest you first consider basic examples and counterexamples when looking for a grasp of the problem: constant functions and indicator functions are a good start. –  Did Oct 3 '12 at 14:59
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Yes, it can: $f(x)=1$ and $g(x)=0$, $x \in \mathbb{R}$.

But it need not be: $f(x)=1$, $g(x)=1/2$, $x \in \mathbb{R}$.

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