# Is there a name for this quantity $\int fg dt- \int f dt \int g dt$?

I was wondering if there is a name for this quantity $\overline{<f,g>} := \int_0^1 fg dt- \int_0^1 f dt \int_0^1 g dt$.

Thanks and regards!

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Never seen this. Should these be three $1$, really? What is the domain of integration? –  1015 Apr 17 '13 at 3:25
$[0,1]$. This quantity is used in covariance of two linear rank statistics. –  Tim Apr 17 '13 at 3:51
Ok. I notice that $f,g$ $L^1$ does not suffice for $fg$ to be $L^1$ as $f=g=1/\sqrt{x}$ shows. And $fg$ is the pointwise product, right? Did you mean: it is "somehow" measuring... In this case, there is no Holder here, as we would $1/p+1/q=1$. –  1015 Apr 17 '13 at 3:56
@julien: you are right. it seems have little to do with Holder's inquality. I updated my post. Yes $fg$ is the pointwise product –  Tim Apr 17 '13 at 4:09

I finally found what this is: that's the covariance (and now I realize you even used the word in your comment above). Indeed: $$\int\left(\left(f-\int f\right)\left(g-\int g\right)\right)=\int\left(fg-\left(\int f\right)g-\left(\int g\right)f+\left(\int f\right)\left(\int g\right)\right)$$ $$\int fg-2\int f\int g+\int f\int g=\int fg-\int f\int g.$$