The K-moment problem: A detailed introduction

We present an expanded expository account of the $K$-moment problem for polynomial algebras over $\R^d$, with special emphasis on compact basic closed semialgebraic sets. The central question is to characterize those linear functionals on $\R[x_1,\dots,x_d]$ which admit representation by integration against a positive Radon measure supported on a prescribed set $K\subseteq\R^d$. We begin with the classical background and with Haviland's formulation of the multidimensional moment problem, then explain how real algebraic geometry enters through quadratic modules, preorderings, and Positivstellens\"atze. The compact case is treated in detail from two complementary perspectives. The geometric route through Schm\"udgen's theorem and the operator-theoretic route through a Gelfand--Naimark--Segal construction and the spectral theorem. We also discuss Putinar's refinement, compare the roles of $T(f)$ and $Q(f)$, and explain how Archimedeanity provides the algebraic shadow of compactness. In order to place the subject in a broader context, we survey determinacy and uniqueness questions, the truncated $K$-moment problem and flat extension phenomena, the relation with sums of squares and Hilbert's seventeenth problem, and the special case of algebraic varieties, where positivity modulo an ideal becomes especially transparent.