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Let $L_p([a,b];\mathbb{R}) = \{f:[a,b]\to\mathbb{R} | f$ is Riemann-integrable $\}$. I want to prove that $$ d_p(f,g) = \left(\int_a^b |f(x)-g(x)|^p dx\right)^{1/p} $$ defines a metric in $L_p([a,b],\mathbb{R}).$

Given $f,g$ integrable I easily see that $d_p(f,g)$ exists and that $d_p(f,f)=0$ and $d_p(f,g) = d_p(g,f)$. I also proved the triangle inequality by proving the Hölder inequality for positive real functions. I'm just stuck in the condition $f\neq g \to d_p(f,g)>0$. For simplicity I'm assuming $f \neq 0$ and trying to prove $\int |f|^p \neq 0$.

I know that $f\ge 0$ implies $\int f \ge 0$ but I don't see how to get the strict inequality. Any help or hint will be appreciated.

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    $\begingroup$ Actually the space should be altered a bit, since $f =0 $ almost everywhere on $[a,b]$ would deduce that $\int_a^b |f|^p = 0$. $\endgroup$ – xbh Sep 5 '18 at 9:24
  • $\begingroup$ Will continuos functions work? $\endgroup$ – AnalyticHarmony Sep 5 '18 at 13:03
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    $\begingroup$ Yes, that would avoid the possibility that $\int |f - g|^p = 0$ but $f \neq g$, since $\int |f|^p = 0$ and $f$ be continuous would deduce that $f \equiv 0$. This claim could be proved by contradiction. $\endgroup$ – xbh Sep 5 '18 at 13:08
  • $\begingroup$ So the usual notation $L_p$ are used for spaces of continuous functions? $\endgroup$ – AnalyticHarmony Sep 5 '18 at 13:09
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    $\begingroup$ Not exactly. Actually $L$ stands for "Lebesgue" integration. Generally $R$ would be used for Riemann integrals. For continuous functions, try $C$, or $C_p$ to indicate the metric. $\endgroup$ – xbh Sep 5 '18 at 13:12
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You can't prove it because it is not true. If$$f(x)=\begin{cases}1&\text{ if }x=a\\0&\text{ otherwise,}\end{cases}$$then $d_p(f,0)=0$, but $f\neq0$. The function $d_p$ is a pseudometric, but not a metric.

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  • $\begingroup$ If I change the set for continuous functions then it'll be a metric? Or is the problem in the defining function per se? $\endgroup$ – AnalyticHarmony Sep 5 '18 at 13:04
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    $\begingroup$ @AnalyticHarmony Yes, if you restrict $d_p$ to the space $\mathcal{C}([a,b])$, then it becomes a metric. $\endgroup$ – José Carlos Santos Sep 5 '18 at 16:31

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