We refine stochastic calculus for symmetric Markov processes without using
time reverse operators. Under some conditions on the jump functions of locally
square integrable martingale additive functionals, we extend Nakao's
divergence-like continuous additive functional of zero energy and the
stochastic integral with respect to it under the law for quasi-everywhere
starting points, which are refinements of the previous results under the law
for almost everywhere starting points. This refinement of stochastic calculus
enables us to establish a generalized Fukushima decomposition for a certain
class of functions locally in the domain of Dirichlet form and a generalized
It\^{o} formula. (With Errata.)Comment: Published in at http://dx.doi.org/10.1214/09-AOP516 and Errata at
http://dx.doi.org/10.1214/11-AOP700 the Annals of Probability
(http://www.imstat.org/aop/) by the Institute of Mathematical Statistics
(http://www.imstat.org