  The analysis of large population economies with incomplete information often
entails the integration of a continuum of random variables. We showcase the
usefulness of the integral notion \`a la Pettis (1938) to study such models. We
present several results on Pettis integrals, including convenient sufficient
conditions for Pettis integrability and Fubini-like exchangeability formulae,
illustrated through a running example. Building on these foundations, we
conduct a unified analysis of Bayesian games with arbitrarily many
heterogeneous agents. We provide a sufficient condition on payoff structures,
under which the equilibrium uniqueness is guaranteed across all signal
structures. Our condition is parsimonious, as it turns out necessary when
strategic interactions are undirected. We further identify the moment
restrictions, imposed on the equilibrium action-state joint distribution, which
have crucial implications for information designer's problem of persuading a
population of strategically interacting agents. To attain these results, we
introduce and develop novel mathematical tools, built on the theory of integral
kernels and reproducing kernel Hilbert spaces in functional analysis.
