This paper is motivated by J. K. Pachl’s work [Pac. J. Math. 82, 519-521 (1979; Zbl 0419.28006)] concerning the Radon measures as functionals on the space \(U_ b(X)\) of all uniformly continuous norm bounded functions on a complete metric space \(X\). Let \(\text{Lip}_ b(X)\) be the space of all norm bounded Lipschitz functions on \(X\) and let \(M_ t(X)\) denote the space of all norm bounded Radon measures on \(X\). Observing the importance of the subspace \(\text{Lip}_ b(X)\) of \(U_ b(X)\), the author first studies the representation theory of some classes of linear functionals on it and gets that \(M_ t(X)\) can be identified with the space of all bounded linear functionals on \(\text{Lip}_ b(X)\) whose restrictions to \(B\) are continuous, where \(B\) is the closed unit ball of \(\text{Lip}_ b(X)\) endowed with the topology of uniform convergence on the compact sets. Next he studies compactness in the weak topology \(w(M_ t(X),\text{Lip}_ b(X))\) and obtains the following: if \(M\subset M_ t(X)\) is a relatively \(w(M_ t(X),\text{Lip}_ b(X))\) compact set with \(| m|(X)=1\) for all \(m\subset \hbox{cl}(M)\) (the closure of \(M\)), then \(w(M_ t(X),C_ b(X))\) coincides with \(w(M_ t(X),\text{Lip}_ b(X))\) on \(M\). Finally, the question whether \(w(M_ t(X),U_ b(X))\) is sequentially complete is considered and it is answered in the following form: \(w(M_ t(X),U_ b(X))\) is sequentially complete if and only if \(X\) is compact.