Harris recurrence is the property that L(x, A) = 1 for all 
and all 
-positive A, where is the stationary distribution.
Harris recurrence is a stronger property than -irreducibility, which
only requires that L(x, A) ;SPMgt; 0 for all x and all -positive A,
although it can be proved that when a chain is -irreducible,
there is a -null set N such that
L(x, A) = 1 does hold for all 
and all -positive A.
(Meyn and Tweedie, 1993, Proposition 9.0.1).
So the point of Harris recurrence is to banish the pathological null set.
It would
be very strange if a Markov chain that is an idealization of a computer
simulation would be -irreducible but not Harris recurrent. If null
sets matter when the computer's real numbers are replaced by those of
real analysis, then the simulation cannot be well described by the theory.
Fortunately, an irreducible Gibbs or Metropolis sampler is always Harris recurrent under very weak conditions (Tierney, 1994, Corollaries 1 and 2) and the same is true of variable-at-a-time Metropolis-Hastings (Chan and Geyer, 1994, Theorem 1). Because of the general measures employed in the MHG algorithm there can be no such simple theorem implying Harris recurrence for general MHG samplers, but the same principles can sometimes be used to check Harris recurrence for them too.
A different way to prove Harris recurrence uses a so-called drift condition. Recall that for any function V
The function V is said to be unbounded off small sets if every
the level set 
is small.
We say a Markov chain satisfies the drift condition for recurrence
if there exists a function 
which is unbounded
off small sets and a small set 
such that
If a chain satisfies the drift condition for recurrence, then it is Harris recurrent (Meyn and Tweedie, 1993, Theorem 9.1.8).