There's a property called "entropy density of ergodic measures" (or variations on that terminology), which states that given an invariant measure μ, you can find a sequence of ergodic measures μn that converges to μ in the weak* topology, and furthermore, the lim inf of the entropies hμn(f) is at least hμ(f). In other words, not only can you approximate an arbitrary invariant measure using ergodic measures, but you can do so without losing any of the entropy of the original measure.
Of course not every system has entropy density, but there are many interesting ones that do. Pfister and Sullivan use this property in a couple papers -- see "Large deviations estimates for dynamical systems without the specification property" (Nonlinearity 18, 2005, 237-261), and "On the topological entropy of saturated sets" (Ergod. Th. & Dynam. Sys. 27, 2007, 929-956). In particular, they show that entropy density follows from something they call the g-almost product property. This latter property is a weaker form of the classical specification property (it's also been called almost specification).
There are many systems known to have specification -- for example, if an Anosov systems, an Axiom A system, a subshift of finite type, or an interval map is topologically mixing, then it has specification, and hence has entropy density. There are also classes of systems that satisfy the g-almost product property (but not specification) and hence have entropy density: β-shifts are one example.