I am reading a paper by Fukushima "On a stochastic calculus related to Dirichlet forms and distorted Brownian motions" and support it by a book "Dirichlet forms and symmetric Markov processes" by Fukushima, Oshima and Takeda.

Let me also remind that a dense subset $\tilde{C}$ of $C_0(X)$ is said to be a core of a Dirichlet form $\mathcal{E}$ if $\tilde{C}$ is $\mathcal{E}_1$-dense in $D[\mathcal{E}]$. A Dirichlet form possessing a core is called regular.

In the paper, there are the following 2 sentences:

"Let $\mathcal{E}$ be a Dirichlet form and $T_t$ the associated semigroup of Markovian symmetric operators on $L^2(X;m)$. If $\mathcal{E}$ is regular, then $T_t$ can be realized as \begin{equation} T_t\,f(x) = \int_X P_t(x, dy) f(y) \end{equation} by a transition function $P_t(x, E)$ on $X$ which is $m$-symmetric in the sense that $\int_XP_t\,f(x)\,g(x)\,m(dx) = \int_X P_tg(x)\,f(x)\,m(dx)$."

I could not find the above result in the book and I do not understand why we need the regularity assumption for that. Namely, we know that the space $C_0(X)$ of continuous functions with compact support is dense in $L^2(X,m)$, so we by Riesz-Markov-Kakutani there exists a transition function $P_t(x,dy)$ such that the above equation holds for $f\in C_0(X)$, and by density we can extend it to $L^2(X;m)$. Please tell me why this is wrong. Thank you very much.