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This paper studies Positivstellens\"atze and moment problems for sets thatare given by universal quantifiers. Let $Q$ be a closed set and let $g =(g_1,...,g_s)$ be a tuple of polynomials in two vector variables $x$ and $y$.Then $K$ is described as the set of all points $x$ such that each $g_j(x, y)\ge 0$ for all $y \in Q$. Fix a measure $\nu$ with $supp(\nu) = Q$, and assumeit satisfies the Carleman condition.  The first main result of the paper is a Positivstellensatz with universalquantifiers: if a polynomial $f(x)$ is positive on $K$, then it belongs to thequadratic module $QM(g,\nu)$ associated to $(g,\nu)$, under the archimedeannessassumption on $QM(g,\nu)$. Here, $QM(g,\nu)$ denotes the quadratic module ofpolynomials in $x$ that can be represented as \[\tau_0(x) + \int\tau_1(x,y)g_1(x, y)\, d\nu(y) + \cdots + \int \tau_s(x,y) g_s(x, y)\, d\nu(y),\] where each $\tau_j$ is a sum of squares polynomial.  Second, necessary and sufficient conditions for a full (or truncated)multisequence to admit a representing measure supported in $K$ are given. Inparticular, the classical flat extension theorem of Curto and Fialkow isgeneralized to truncated moment problems on such a set $K$. Finally,applications of these results for solving semi-infinite optimization problemsare presented.

Positivstellens\"atze and Moment problems with Universal Quantifiers