1. ## L2 space

Let $f_n,f,g \in L^2(X,F,\mu)$ and $|f_n|\leq g$ for all $n$.
Show that $f_n\rightarrow f$ in $L^2$ $iff$ $f_n \rightarrow f$ in measure and $\int f_n^2d\mu\rightarrow \int f^2d\mu$.
i could show that if $f_n\rightarrow f$ in $L^2$ , $\int f_n^2d\mu\rightarrow \int f^2d\mu$.
but im stuck on showing if $f_n\rightarrow f$ in $L^2$, $f_n \rightarrow f$ in measure. please help me on this. thank you.

2. Originally Posted by GTO
Let $f_n,f,g \in L^2(X,F,\mu)$ and $|f_n|\leq g$ for all $n$.
Show that $f_n\rightarrow f$ in $L^2$ $iff$ $f_n \rightarrow f$ in measure and $\int f_n^2d\mu\rightarrow \int f^2d\mu$.
i could show that if $f_n\rightarrow f$ in $L^2$ , $\int f_n^2d\mu\rightarrow \int f^2d\mu$.
but im stuck on showing if $f_n\rightarrow f$ in $L^2$, $f_n \rightarrow f$ in measure. please help me on this. thank you.
Since $f_n \rightarrow f$ in $L^2$, we have $\int_X |f-f_n|^2 d\mu \rightarrow 0$. Let $E=\{x:|f-f_n|>\epsilon\}$ and $D= \{x:|f-f_n|<\epsilon\}$. Then $\int_X |f-f_n|^2 d\mu=\int_E |f-f_n|^2 d\mu +\int_D |f-f_2|^2 d\mu \rightarrow 0$. So $\int_E |f-f_n|^2 d\mu\rightarrow 0$.

i got this far but i dont know how i can show $|f-f_n|<\infty$. i am thinking that the given assumption $|f_n| can be used here but i dont know how i can use it. please help me .

3. Originally Posted by GTO
Since $f_n \rightarrow f$ in $L^2$, we have $\int_X |f-f_n|^2 d\mu \rightarrow 0$. Let $E=\{x:|f-f_n|>\epsilon\}$ and $D= \{x:|f-f_n|<\epsilon\}$. Then $\int_X |f-f_n|^2 d\mu=\int_E |f-f_n|^2 d\mu +\int_D |f-f_2|^2 d\mu \rightarrow 0$. So $\int_E |f-f_n|^2 d\mu\rightarrow 0$.

i got this far but i dont know how i can show $|f-f_n|<\infty$. i am thinking that the given assumption $|f_n| can be used here but i dont know how i can use it. please help me .
Can i assume that $f_n$ is bounded since $\int |f_n|^2 <\infty$?

4. Originally Posted by GTO
Since $f_n \rightarrow f$ in $L^2$, we have $\int_X |f-f_n|^2 d\mu \rightarrow 0$. Let $E=\{x:|f-f_n|>\epsilon\}$ and $D= \{x:|f-f_n|<\epsilon\}$. Then $\int_X |f-f_n|^2 d\mu=\int_E |f-f_n|^2 d\mu +\int_D |f-f_2|^2 d\mu \rightarrow 0$. So $\int_E |f-f_n|^2 d\mu\rightarrow 0$.

i got this far but i dont know how i can show $|f-f_n|<\infty$. i am thinking that the given assumption $|f_n| can be used here but i dont know how i can use it. please help me .
i am not really sure but since $\int |f-f_n|^2 < \epsilon$ for all $n>N$, $|f-f_n|<\infty$. otherwise this integral will not go to 0. can someone tell me if it is correct?