An $L^1$ functional from a space $X$ to $\mathbb{R}$ is an $\mu$-measurable function such that $$ \int_{X} |f|\,d\mu < \infty. $$
My question is if suppose I pick one function $f_1$ from $L^1$ so we will have $\int_{X}|f|d\mu= C_1(say)< \infty$, similarly for some $f_2\in L^1$ we have $\int_{X}|f_2|d\mu=C_2(say)<\infty$ so if I collect all function from $L^1$ then there integral will be finite say it is some number $C_i$ where $i\in I$(Indexing set). So if I find the $sup_{i}\int_{X}|f_i|d\mu$ will it be finite?? I mean this $sup_{i}\int_{X}|f_{i}|d\mu=C<\infty$ where $C=max\{C_1,C_2,.....\}$? If it is true then can someone give example of such a family of functions.