Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L_2(\mu)$ with $\mathrm{supp} \, \gamma = L_2(\mu)$ and existing first moment. Then
$$ f \mapsto \|f\| := \int |\langle f,g\rangle|\,\gamma(\mathrm{d}g) $$
is a norm on $L_2(\mu).$ Does this norm have a name? Can anyone give me some references where such normed vector spaces were studied?