Simultaneous diagonalization of self-adjoint operators on Hilbert space

Apply the SNAG (Stone-Naimark-Godement-Ambrose) theorem to the unitary group generated by these operators.

Another way to do it is to consider the bounded operators $(T_i + iI)^{-1}$, check that these are normal and commute so they can be simultaneously represented as multiplication operators on an $L^2$ space, and then untransform to represent the original $T_i$ as unbounded multiplication operators.

There is a very nice treatment of this question in the textbook by Konrad Schmüdgen "Unbounded Self-adjoint Operators on Hilbert Space", see in particular Theorem 5.23 p. 103.

While there is no problem constructing a unique product measure $\mu\otimes\nu$ for $\sigma$-finite measure spaces $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N},\nu)$ the analogous question for projection-valued measures encounters a snag unless $X$ and $Y$ are sufficiently nice like $\mathbb{R}^n$. See the remark after Theorem 4.10 in the same book and the reference to the article by Birman, Vershik and Solomjak.