  We generalize the notion of joint measurability to continuous variable
systems by extending a recently introduced compression algorithm of quantum
measurements to this realm. The extension results in a property that asks for
the minimal dimensional quantum system required for representing a given set of
quantum measurements. To illustrate the concept, we show that the canonical
pair of position and momentum is completely incompressible. We translate the
concept of measurement compression to the realm of quantum correlations, where
it results in a generalisation of continuous variable quantum steering. In
contrast to the steering scenario, which detects entanglement, the
generalisation detects the dimensionality of entanglement. We illustrate the
bridge between the concepts by showing that an analogue of the original EPR
argument is genuinely infinite-dimensional with respect to our figure of merit,
and that a fundamental discrete variable result on preparability of unsteerable
state assemblages with separable states does not directly carry over to the
continuous variable setting. We further prove a representation result for
partially entanglement breaking channels that can be of independent interest.
