  We present a generalization of the induced matching theorem and use it to
prove a generalization of the algebraic stability theorem for
$\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via
numerous examples, we show how the generalized algebraic stability theorem
enables the computation of rigorous error bounds in the space of persistence
diagrams that go beyond the typical formulation in terms of bottleneck (or log
bottleneck) distance.
