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Homework Statement
Suppose f \in L^1((0,\infty), dm). Prove that
\[ L := \lim_{t\to\infty} \frac{1}{t}\int_0^t x\, f(x)\, dm(x) = 0. \]
Homework Equations
dm(x) represents integration with respect to Lebesgue measure.
The Attempt at a Solution
I know how to do this if f is in Lp and not just L1 as long as the "x" and "1/t" aren't there,.. I tried a similar approach here but can't do any better than this:
<br /> \begin{align*}<br /> \left|\int_0^t xf(x)\, dm(x)\right| &\le \int_0^\infty |xf(x) \chi_{[0,t]}(x)|\, dm(x)\\<br /> &\le \|f\|_1 \cdot \|x \chi_{[0,t]}(x)\|_\infty\\<br /> &= \|f\|_1 \cdot t.<br /> \end{align*}<br />
The last inequality is Hölder with p=1 and q = \infty.
But then L \le \lim_{t\to\infty} \|f\|_1, which only shows that L is finite but not necessarily zero. Thanks in advance!
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