  Submodularity is a key property in discrete optimization. Submodularity has
been widely used for analyzing the greedy algorithm to give performance bounds
and providing insight into the construction of valid inequalities for
mixed-integer programs. In recent years, researchers started to study
approximate submodularity, with a primary focus on providing performance bounds
for iterative approaches. In this paper, we study approximate submodularity
from a different perspective in order to broaden its use cases in discrete
optimization. We define metrics that quantify approximate submodularity, which
we then use to derive new properties about both approximate submodularity
preservation and the well-known Lov\'{a}sz extension for set functions. We also
show that previous analyses of mixed-integer sets, such as the submodular
knapsack polytope, can be extended to the approximate submodularity setting.
Our work demonstrates that one may generalize many of the analytical tools used
in submodular optimization into the approximate submodularity context.
