Let $(f_j)_{j\in\mathbb{N}}$ be a sequence of real functions in $L^p(X,\mathcal{A},\mu)$, where $p\geq1$. If we know that $$\lim\limits_{j\rightarrow\infty}\int_X|f_j|^pd\mu=\int_X|f|^pd\mu\space\space\space(\dagger),$$ for some measurable function $f$ and $f_j\rightarrow f$, in measure.
My question is when is $f\in L^p(X,\mathcal{A},\mu)$, i.e. $\int_X|f|^pd\mu<\infty$? If for any $j\in\mathbb{N}$ we have $|f_j|\leq g$ for some $g\in L^1$ we can use the dominated convergence theorem, but we don't know if that is the case in this situation, or does $(\dagger)$ imply it(if so how?). Please would appreciate any answers and feedback to my question. Thanks.