Doubt about definition of convergence in $L^p$.

Question

I have a question concerning the definition of convergence in $L^p$.

To say $(f_i) \rightarrow f$ in $L^p$, do we need $(f_i)\subset L^p$ and $f \in L^p$? This would allow the integral $ \int _ \Omega |f_i-f|^p d\mu$ to always be defined. However, for $ g \notin L^p$, the constant sequence $(g)$ would not converge to $g$ itself in $L^p$.

Thank you very much

Answer 1

Yes, convergence in a space requires the limit to be in that space.