  When you measure an observable, A, in Quantum Mechanics, the state of the
system changes. This, in turn, affects the quantum-mechanical uncertainty in
some non-commuting observable, B. The standard Uncertainty Relation puts a
lower bound on the uncertainty of B in the initial state. What is relevant for
a subsequent measurement of B, however, is the uncertainty of B in the
post-measurement state. We re-examine this problem, both in the case where A
has a pure point spectrum and in the case where A has a continuous spectrum. In
the latter case, the need to include a finite detector resolution, as part of
what it means to measure such an observable, has dramatic implications for the
result of successive measurements. Ozawa proposed an inequality satisfied in
the case of successive measurements. Among our results, we show that his
inequality is ineffective (can never come close to being saturated). For the
cases of interest, we compute a sharper lower bound.
