Exercise 12, § 8

Let $T_i$ ($1 \leq i \leq n$) be $n$ locally compact spaces, $\mu_i$ a positive measure on $T_i$ for $1 \leq i \leq n$. For each index $i$, denote by $E_i$ the product $\prod_{j \neq i} T_j$; let $f_i$ be a function $\geq 0$ on $T = \prod_{i=1}^n T_i$, measurable for $\mu = \bigotimes_{i=1}^n \mu_i$ and not dependent on $t_i$; show that if, for $1 \leq k \leq n$, the function $f_k^{n-1}$ is integrable for the measure $\mu_1 \otimes \cdots \otimes \mu_{k-1} \otimes \mu_{k+1} \otimes \cdots \otimes \mu_n$, then $f_1 f_2 \cdots f_n$ is $\mu$-integrable, and

where, for each index $k$, one sets $J_k = \int f_k^{n-1} d\mu_1 \cdots d\mu_{k-1} d\mu_{k+1} \cdots d\mu_n$ (proceed by induction on $n$, applying the Lebesgue–Fubini theorem and Hölder’s inequality).

Deduce from this that if $A$ is a $\mu$-measurable subset of $T$, $A_i$ its projection on $E_i$, and if $A_i$ is integrable and of measure $m_i$ (for the measure $\bigotimes_{j \neq i} \mu_j$ on $E_i$), then $A$ is $\mu$-integrable and

Examine the case of equality in these two inequalities.

Generalize to the case where, instead of considering the $n$ products of $n-1$ of the $T_i$, one considers the $\binom{n}{p}$ products of $p$ of the $T_i$, and where one integrates over $T$ a product of $\binom{n}{p}$ functions $\geq 0$ each of which depends on only $p$ of the variables $t_i$. For example, if $F_{ij} = T_i \times T_j$ ($i < j$) and if $f_{ij}$ depends only on the variables $t_i$ and $t_j$, show that