Additivity of the variance is a characteristic property of the Hilbert space \(L_ 2(\Omega,{\mathfrak A},\mu)\) (Q788956)

Let (\(\Omega\),\({\mathfrak A},\mu)\) be a measure space with finitely continuous measure \(\mu\), E be a rearrangement-invariant Banach function space on \(\Omega\). If x is a random variable and \(m(x)=\int x(\omega)d\mu(\omega)\) its mean value, we define \(\delta(x)=\| x- m(x)\|_ E\). For \(E=L_ 2(\Omega,{\mathfrak A},\mu)\) we have \(\delta^ 2(x)=D(x)\), where D(x) is the usual dispersion of x. In this case \(D(x+y)=D(x)+D(y)\) iff the random variables x and y are uncorrelated. The aim of this work is to show, that the validity of equality \(\delta^ 2(x+y)=\delta^ 2(x)+\delta^ 2(y)\) for all independent x,\(y\in E\) is a characteristic property of \(L^ 2(\Omega,{\mathfrak A},\mu)\) in some class of rearrangement-invariant spaces E.