Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $f\_{n,j}$ be a doubly indexed sequence of positive functions in $L^p(\Omega),$ $1<p<\infty.$ Suppose $f\_{n,j}$ converges pointwise a.e. to $f\_n\in L^p$ and also in $\|.\|\_p.$ Denote $f\_{n}^\*(x)=\sup\_{j\geq 1} f\_{n,j}(x)$ and $f^\*(x)=\sup\_{n\geq 1}f\_n(x).$ Assume $\sup\|f\_n\|\_p<\infty.$ Is it true that $\|f^\*\|\_{p}\leq\sup\_{n\geq 1}\|f\_n^\*\|\_p$ ?



Limit of doubly indexed functions
The answer is no. E.g., suppose that $\Omega=\mathbb R$ with the Lebesgue measure $\mu$ over $\mathbb R$ and $f\_{n,j}=f\_n=1\_{[n,n+1)}$ for natural $n,j$. Then $f\_n^\*=f\_n=1\_{[n,n+1)}$ and $f^\*=1\_{[1,\infty)}$. So, 
$$\|f^\*\|\_p=\infty>1=\sup\_n\|f\_n^\*\|\_p.
$$


