Every family of functions that is total in $\mathcal{C}_0(\mathbb{R}^n)$ works, by linearity and density.

The notation I know is $\mu K$, not $K \mu$.

Is W a (standard) one-dimensional Brownian motion? If yes, it would be clearer to give the equation of $L$ in terms of $t$ (time variable) and $x$ (space variable), instead of $t$ and $x$. Or is it a two dimensional Brownian motion?

Distribution of local times at a fixed times are not simple. An approach is to work at an exponential independent time with parameter $\lambda$ (at this time, Ray's theorem or first Ray-Knight theorem applies), and then to invert a Laplace transform to deduce the distribution at a given fixed time. Yet, this strategy may be difficult to follow.

Yes. Given $\epsilon>0$, Integrability and equidistribution yields the convergence of the series $\sum_n P[|X_n|>\epsilon n]$. Then, independence and Borel Cantelli lemma shows that almost surely $|X_n| \le n \epsilon$ eventually. Therefore, $X_n/n \to 0$ almost surely, and $X^*_n/n \to 0$ almost surely.

@MichaelGreinecker No. Take $x=r$ and $K=[r,r+1]$. Then for every $\delta \in (0,1)$, the sup on $[r,r+\delta]$ equals $1$, whereas the esssup equals $0$.

@tsnao When $t \in [0,1]$, $t(1-t) \le 1/4$.

Integer partitions with multiplicities at most $k$ is a bit long, but clear, and I do not see how to do better.