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Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-integrable functions from X to Y.

Let $\operatorname{Lip}_c(X,Y)$ denote the set of Lipschitz functions from $X$ to $Y$ with closed and bounded support.


If $1 \le p< \infty$ then is $\operatorname{Lip}_c(X,Y)$ dense in $L^p(X,Y;\mu)$?


If not, then what if we additionally assume that X and Y are Hilbert spaces?

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    $\begingroup$ I think it is true without separability assumptions on $X$ and $Y$. It is sufficient to approximate a function $v\chi_S$ for $v\in Y$ and $S$ Borel measurable in $X$, so one is reduced to the scalar case. For $S$ open it is OK via distance function. From open to Borel it follows e.g. via the standard monotone class argument $\endgroup$ Sep 2, 2022 at 11:43
  • $\begingroup$ @PietroMajer Could I ask for more details as an answer? $\endgroup$
    – Wilson
    Sep 5, 2022 at 22:05

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