For a hidden Markov process $\{x_k\}_{k\in {\mathbb N}_0}$ with observations $\{y_k\}_{k\in {\mathbb N}_0}$, the Chapman-Kolmogorov equation is
$$p(x_k|y_0,\ldots,y_{k-1}) = \int_{{\mathcal X}_{k-1}} p(x_k|x_{k-1}) p(x_{k-1}|y_0,\ldots,y_{k-1}) {\mathrm d}x_{k-1}.$$
This form is very similar to the law of total probability. I think they are equivalent, am I right? Thanks!