Consider $R$-valued functions in $L^p(X,\Sigma,\lambda)$, where $X=X^1\times X^2$, $\Sigma=\Sigma^1\times \Sigma^2$ and $\lambda=\lambda^1\times \lambda^2$
For given $i$, does the subsapce
$M=\{f\in L^p(X ,\Sigma,\lambda): f(\cdot,x^{-i})=f(\cdot,\tilde{x}^{-i}), \forall x,\tilde{x}\in X \}$ has the same property as $L^p(X^i ,\Sigma^i,\lambda^i)$?
(i) Is $M$ complete?
(ii) $\{ f\in M| f=\sum_{\alpha_{n}} \chi_{E^i\times T^{-i}}, E^i\in \Sigma^i\}$ is dense in $M$?
(iii) If $M^1\subset M$ is convex and compact and $g\notin M^1$, then there exists $f\in M$ such that $\langle f,g \rangle>\sup_{f'\in M^1} \langle f',g \rangle$?
Thanks.