Let Pt be a submarkov semigroup on a Lusin topological state space E with Borel field E. We call a positive sigma-finite measure m on E an eigenmeasure if mPt = e-ct m for some real number c. We call a positive E-measurable function f an eigenfunction if ptf = e-ctf for some real number c. In each case, we call c the eigenvalue of either the eigenmeasure or the eigenfunction. Eigenmeasures are also known by the name quasi-stationary distributions in the Markov chain literature: see [5], [17], [18].