  In this paper, we introduce the notion of $L^p$-Green-tight measures of
$L^p$-Kato class in the framework of symmetric Markov processes. The class of
$L^p$-Green-tight measures of $L^p$-Kato class is defined by the $p$-th power
of resolvent kernels. We first prove that under the $L^p$-Green tightness of
the measure $\mu$, the embedding of extended Dirichlet space into
$L^{2p}(E;\mu)$ is compact under the absolute continuity condition for
transient Markov processes, which is an extension of recent seminal work by
Takeda. Secondly, we prove the coincidence between two classes of
$L^p$-Green-tightness, one is originally introduced by Zhao, and another one is
invented by Chen. Finally, we prove that our class of $L^p$-Green-tight
measures of $L^p$-Kato class coincides with the class of $L^p$-Green tight
measures of Kato class in terms of Green kernel under the global heat kernel
estimates. We apply our results to $d$-dimensional Brownian motion
androtationally symmetric relativistic $\alpha$-stable processes on
$\mathbb{R}^d$.
