What you "need to show" is of course false in general. E.g., suppose that $d=1$ and $\varphi=1_{[0,1]}$. Then your set of measures is [not compact even in the topology of weak convergence][1], since the the sequence of the uniform distributions over the intervals $[n,n+1]$ for natural $n$ is not tight. 


  [1]: https://en.wikipedia.org/wiki/Prokhorov%27s_theorem