In my lecture notes it says:
Let H be a $k$-Hopf algebra. Let M be a left H-module. The invariants of H on M are defined as the $k$-vector subspace
$M^{H}$:= {m $\in$ M | h.m=$\epsilon$(h)m for all h $\in$ H} of M.
Why does the counit $\epsilon$ appear?
I am aware that this is not a particularly precise question but maybe answers will be insightful still.