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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.

user152874
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    By your assumptions this is the case iff the limit $\lim \int |f_j|^p dx $ is finite. But I don't really think that this is your question. Note that you did not say anything about pointwise convergence. Thus your remark about dominated convergence is not really applicable. – PhoemueX Jun 14 '14 at 13:14
  • Thanks, for the feedback. Sorry, my question is under what conditions on $f$ can we say that $\lim\limits_j\int|f_j|^pd\mu<\infty$. I did not mention convergence (which was a mistake) as I want to know what mode of convergence do we need, so must we have $f_j\rightarrow f$ pointwise, or can we weaken it to say that $f_j\rightarrow f$ in measure. In both cases we can then use the dominated convergence (with the addition that the space is $\sigma$-finite if we have convergence in measure) but only if we have a function that dominates the sequence, but if we don't or does $(\dagger)$ imply we do? – user152874 Jun 14 '14 at 14:27

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