_{L^q({\mathbf R^{d}})}},\end{aligned}$$ respectively.

More generally, let $${{\boldsymbol p}}=(p_1,\dots , p_d)\in (0,\infty ]^d,
\quad {\boldsymbol q}=(q_1,\dots , q_d)\in (0,\infty ]^d,$$ $\operatorname {S}_d$ be the set of permutations on $\{ 1,\dots ,d\}$, ${{\boldsymbol p}}\in
(0,\infty ]^d$, $\omega \in {\mathscr P}_E({\mathbf R^{d}})$, and let $\sigma
\in \operatorname {S}_d$. Moreover, let $\Omega _j\subseteq
\mathbf R$ be Borel-sets, $\mu _j$ be positive Borel measures on $\Omega _j$, $j=1,\dots ,d$, and let $\Omega =\Omega _1\times \cdots \times \Omega _d$ and $\mu = \mu _1\otimes \cdots \otimes \mu _d$. For every measurable and complex-valued function $f$ on $\Omega$, let $g_{j,\omega ,\mu}$, $j=1,\dots ,d-1$, be defined inductively by $$\begin{aligned}
g_{0,\omega ,\mu}(x_1,\dots ,x_d)
&\equiv |f (x_{\sigma ^{-1}(1)},\dots ,x_{\sigma ^{-1}(d)})
\omega (x_{\sigma ^{-1}(1)},\dots ,x_{\sigma ^{-1}(d)})|,
\\[1ex]
g_{k,\omega ,\mu}(x_{k+1},\dots ,x_d) &\equiv {\Vert g_{k-1,\omega ,\mu}({\, \cdot \, },
x_{k+1},\dots ,x_d) \Vert _{L^{p_k}(\mu _k)}},
\quad k=1,\dots ,d-1 ,
\intertext{and let }
{\Vert f\Vert _{L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu)}} &\equiv
{\Vert g_{d-1,\omega ,\mu}\Vert _{L^{p_d}(\mu _d)}}.\end{aligned}$$ The mixed quasi-norm space $L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu)$ of Lebesgue type is defined as the set of all $\mu$-measurable functions $f$ such that ${\Vert f\Vert _{L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu )}}<\infty$.

In the sequel we have $\Omega ={\mathbf R^{d}}$ and $d\mu = dx$, or $\Omega =\Lambda$ and $\mu (j)=1$ when $j 