  In this short note we show how the Generalised Uncertainty Principle (GUP)
and the Extended Uncertainty Principle (EUP), two of the most common
generalised uncertainty relations proposed in the quantum gravity literature,
can be derived within the context of canonical quantum theory, without the need
for modified commutation relations. A GUP-type relation naturally emerges when
the standard position operator is replaced by an appropriate Positive Operator
Valued Measure (POVM), representing a finite-accuracy measurement that
localises the quantum wave packet to within a spatial region $\sigma_g > 0$.
This length scale is the standard deviation of the envelope function, $g$, that
defines the POVM elements. Similarly, an EUP-type relation emerges when the
standard momentum operator is replaced by a POVM that localises the wave packet
to within a region $\tilde{\sigma}_g > 0$ in momentum space. The usual GUP and
EUP are recovered by setting $\sigma_g \simeq \sqrt{\hbar G/c^3}$, the Planck
length, and $\tilde{\sigma}_g \simeq \hbar\sqrt{\Lambda/3}$, where $\Lambda$ is
the cosmological constant. Crucially, the canonical Hamiltonian and commutation
relations, and, hence, the canonical Schr{\" o}dinger and Heisenberg equations,
remain unchanged. This demonstrates that GUP and EUP phenomenology can be
obtained without modified commutators, which are known to lead to various
pathologies, including violation of the equivalence principle, violation of
Lorentz invariance in the relativistic limit, the reference frame-dependence of
the `minimum' length, and the so-called soccer ball problem for multi-particle
states.
