This refers to the following problem. Given a sequence of numbers, when are they the moments of some measure (or distribution if you like to normalize to the case that is known as probability.) That is, given a sequence $(a_n)^{\infty}_{n=0}$ when is there a measure such that $a_n = \int x^n d\mu{x}.$ To solve this problem, we introduce some basic modern analysis, specifically Hilbert space techniques, which were invented early in the last century to solve problems like the Hamburger moment problem.