  We axiomatically introduce risk-consistent conditional systemic risk measures
defined on multidimensional risks. This class consists of those conditional
systemic risk measures which can be decomposed into a state-wise conditional
aggregation and a univariate conditional risk measure. Our studies extend known
results for unconditional risk measures on finite state spaces. We argue in
favor of a conditional framework on general probability spaces for assessing
systemic risk. Mathematically, the problem reduces to selecting a realization
of a random field with suitable properties. Moreover, our approach covers many
prominent examples of systemic risk measures from the literature and used in
practice.
