Build a sequence $u_k $

[Camillo]

Build $u_k in C^0(RR) nn L^1(RR) $ converging in $L^1(RR)$ to the characteristic function of $[0,1] $ and verifying :$0 <= u_k (x)<= 1 $ a.e. in $RR, AA k$.
In which further $L^p $ space will the $ u_k(x) $ sequence converge , according to theory ?
Verify directly on the $ u_k $ found.

[Camillo]

In addition to Luca and El giovo solutions I indicate mine herebelow :

$u_k = kx $ for $ 0<= x<=1/k $
$u_k = 1 $ for $ 1/k $u_k = -k(x-1) $ for $ 1- 1/k

Cita 0 0

Segnala

[Luca.Lussardi]

An affine function, by definition, has a straight-line as graph. You have to connect the point $(1,1)$ with the point $(1+1/n,0)$ in view to obtain the graph of the function $u_k$ (the same for the points $(0,1)$ and $(-1/n,0)$).

[Camillo]

"Luca.Lussardi":For instance $u_k=1$ on $[0,1]$, $u_k=0$ on $(-\infty,-1/n) \cup [1+1/n,+\infty)$; otherwise $u_k$ is an affine function in such a way that $u_k \in C^0(\RR)$. Then $u_k \to \chi_{(0,1)}$ in $L^1(\RR)$ and it has compact support uniformly, then it belongs to any $L^p(\RR)$.

Remark: $u_k$ does not converge to $\chi_{(0,1)}$ in $L^\infty(\RR)$.

Which is exactly the meaning of affine function ? and how can you conclude that $u_k inC^0(RR) $ ? .

[elgiovo]

What about this?

$u_k(x)=mbox(exp){-(2x-1)^(2k)}$

In this case $u_k$ belongs to any $L^p(RR)$.

[Luca.Lussardi]

For instance $u_k=1$ on $[0,1]$, $u_k=0$ on $(-\infty,-1/n) \cup [1+1/n,+\infty)$; otherwise $u_k$ is an affine function in such a way that $u_k \in C^0(\RR)$. Then $u_k \to \chi_{(0,1)}$ in $L^1(\RR)$ and it has compact support uniformly, then it belongs to any $L^p(\RR)$.

Remark: $u_k$ does not converge to $\chi_{(0,1)}$ in $L^\infty(\RR)$.