Local constant interpolation in $L^1$

Local constant interpolation in $L^1$

I really hope that anybody of you can help me with the following question:
Consider the set $U\subseteq L^1([0,1])$ of non-negative integrable
functions with unit mass, i.e. $u\geq 0$, $\int_0^1 u\,dx = 1$ for all
$u\in U$. Further define for any $K\in\mathbb{N}$, $\delta:=\frac{1}{K}$,
the decomposition of $[0,1]$ by the intervals
$I_k:=(k\delta,(k+1)\delta]$, $k=0,\ldots, K-1$.
Moreover define the local constant and isometric interpolation map
\begin{align*} P_{\delta}(u):\begin{cases} L^1([0,1]) \rightarrow
L^1([0,1]) \\ u \mapsto \sum_{k=0}^{K-1} u_k \chi_{I_k}, \qquad with\qquad
u_k:=\frac{1}{\delta}\int_{I_k} u\, dx \end{cases}\end{align*} where
$\chi_A$ standy for the indicator function for any subset $A$ of $[0,1]$.
Now, my question is as follows: Is it possible to proof uniform
convergence on $U$ of $P_{\delta}$ towards the identity in $L^1$, i.e.
\begin{align*} \forall \varepsilon>0\,\exists\bar\delta:
\|P_{\delta}(u)-u\|_{L^1([0,1])}
<\varepsilon,\,\forall\delta\in(0,\bar\delta),\forall u\in U \quad ???
\end{align*} I suppose that one needs more restrictions on $U$, maybe
$\|u\|_{L^{\infty}} \leq C$ for any $C>0$, but to be honest, i have no
idea. My problem is that if you look for a proof of
$\|P_{\delta}(u)-u\|_{L^1([0,1])}\rightarrow 0$ for only one $u\in
L^1([0,1])$, one always uses the existence of stepfunctions $\varphi_n$,
which converges towards $u$ in $L^1$, thus i don't get any inequality like
$\|P_{\delta}(u)-u\|_{L^1([0,1])}\leq C_u\delta$ with $u$-dependent
constant $C_u$, which would be helpful to find a uniform constant
independently of $u$.
If anybody has ideas, i'm happy for any advise. Mabye someone knows a
reference where i can find a convergence proof for a single $u$ without
just using thightness of the set of stepsfunctions in $L^1$... even this
would be very helpful!
Thanks!!