  We show that risk-constrained functional optimization problems with general
integrable nonconvex instantaneous reward/constraint functions exhibit strong
duality, regardless of nonconvexity. We consider risk constraints featuring
convex and positively homogeneous risk measures admitting dual representations
with bounded risk envelopes, generalizing expectations. Popular risk measures
supported within our setting include the conditional value-at-risk (CVaR), the
mean-absolute deviation (MAD, including the non-monotone case), certain
distributionally robust representations and more generally all real-valued
coherent risk measures on the space $L_1$. We highlight the usefulness of our
results by further discussing various generalizations of our base model,
extensions for risk measures supported on $L_{p>1}$, implications in the
context of mean-risk tradeoff models, as well as more specific applications in
wireless systems resource allocation, and supervised constrained learning. Our
core proof technique appears to be new and relies on risk conjugate duality in
tandem with J. J. Uhl's weak extension of A. A. Lyapunov's convexity theorem
for vector measures taking values in general infinite-dimensional Banach
spaces.
