Below we use Bochner measurability and Bochner integral. Let
- $(Y, d)$ be a separable metric space,
- $\mathcal B$ Borel $\sigma$-algebra of $Y$,
- $\nu$ a $\sigma$-finite Borel measure on $Y$,
- $(E, |\cdot|)$ a Banach space,
- $L^0(Y) := L^0(Y, \nu, E)$ the space of $\nu$-measurable functions from $Y$ to $E$,
- $p \in [1, \infty)$,
- $L^p (Y) := L^p(Y, \nu, E)$ the space of $p$-integrable functions from $Y$ to $E$.
Let $(y_n)$ be a countable dense subset of $Y$. For $r>0$ and $f \in L^0(Y)$, we define $$ \begin{align*} \| f \| &:= \sup_{y \in Y} \|1_{B(y, 1)} f\|_{L^p}, \\ [f]_r &:= \sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}, \end{align*} $$ where $B(y, r)$ is the open ball centered at $y$ with radius $r$.
Is there $r>0$ such that the norms $\| \cdot \|$ and $[\cdot]_s$ are equivalent?
The answer is affirmative in the special case where $E=\mathbb R$ and $\nu$ is the Lebesgue measure of $Y = \mathbb R^d$.
Thank you so much for your elaboration!
