Limit of doubly indexed functions

Question

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 $\|f^*\|_{p}\leq\sup_{n\geq 1}\|f_n^*\|_p$ ?

Answer 1

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, $$\|f^*\|_p=\infty>1=\sup_n\|f_n^*\|_p. $$