Say $(S,\Sigma,\mu)$ and $(\Omega,\mathcal F,P)$ are measure spaces and $X : S \times \Omega \rightarrow \mathbb R$ a measurable function. The Wikipedia article on the Minkowski inequality (https://en.wikipedia.org/wiki/Minkowski_inequality#Minkowski's_integral_inequality) defines the quantity $$ || X ||_{p,q} = \big( \int_S \big(\int_\Omega |X|^q dP\big)^{\frac p q} d\mu \big)^{\frac 1 p} $$ and the function space $\mathcal L_{p,q}$, in which it is finite. The article states that these function spaces have been studied, but gives no source. I also wasn't able to find any other resources.
I'm interested because I'm investigating the setting where $(\Omega,\mathcal F,P)$ is a probability triple and $X$ a stochastic process, in which the pointwise expectation $$ \mathbb E[X] : S \rightarrow \overline{\mathbb R}, s \mapsto \mathbb E[X(s)] $$ satisfies $\mathbb E[|X|^q]^{1/q} \in \mathcal L_p$, i.e. $$ || X ||_{p,q} =\big( \int_S \big( \int_\Omega |X|^q dP \big)^{\frac p q} d\mu \big)^{\frac 1 p} = ||\mathbb E[|X|^q]^{\frac 1 q} ||_p < \infty $$ and hence $X \in \mathcal L_{p,q}$.