Give an example of a sequence of functions ($ f_n \epsilon L^1 ([0,1])$ for all $n=1,2,3,...$) and a function $g\epsilon L^1 ([0,1])$ with the following properties.
a. $ f_n (x) \rightarrow g(x) $ for almost all $x\epsilon [0,1]$
b. $\int_{[0,1]} |f_n|d \lambda = 2 $ for every $n=1,2,3... $
c. $\int_{[0,1]} |g|d \lambda = 1 $
I am having difficulty even picturing the answer.
Take $g=1$ and $f_{n}=1+(n+1)x^{n}$, then $\|f_{n}\|_{L^{1}[0,1]}=2$ and $f_{n}(x)\rightarrow 1$ for all $x\in[0,1)$.