  For any pair of bounded observables $A$ and $B$ with pure point spectra, we
construct an associated "joint observable" which gives rise to a notion of a
joint (projective) measurement of $A$ and $B$, and which conforms to the
intuition that one can measure non-commuting observables simultaneously,
provided one is willing to give up arbitrary precision. As an application, we
show how our notion of a joint observable naturally allows for a construction
of a "functional calculus," so that for any pair of observables $A$ and $B$ as
above, and any (Borel measurable) function $f: \mathbb{R}^2 \rightarrow
\mathbb{R}$, a new "generalized observable" $f(A,B)$ is obtained. Moreover, we
show that this new functional calculus has some rather remarkable properties.
