  A classical fact in ergodic theory is that ergodicity is equivalent to almost
everywhere divergence of ergodic sums of all nonnegative integrable functions
which are not identically zero. We show two methods, one in the measure
preserving case and one in the nonsingular case, which enable one to prove this
criteria by checking it on a dense collection of functions and then extending
it to all nonnegative functions. The first method is then used in a new proof
of a folklore criterion for ergodicity of Poisson suspensions which does not
make any reference to Fock spaces. The second method which involves the double
tail relation is used to show that a large class of nonsingular Bernoulli and
inhomogeneous Markov shifts are ergodic if and only if they are conservative.
In the last section we discuss an extension of the Bernoulli shift result to
other countable groups including $\mathbb{Z}^{d},\ d\geq 2$ and discrete
Heisenberg groups.
