Mosses are encouraged by lower pH and discouraged by higher pH. It would be ineffective to apply lime (that is alkaline) to an already alkaline soil. In such a case, there would be other factors involved that would need to be addressed. Fortunately, native soils in northern Wisconsin are almost always acidic, and raising the pH (with lime) is likely the best single method to eradicated moss.
I don't know about the southern lawn grasses, but the lawn grasses in the northern half of the US are quite pH tolerant. When you lime a lawn, you are changing the natural environment to encourage certain flora and fauna, and discourage other flora and fauna. Applying lime to an acidic soil will have a greater effect than applying it to an already alkaline soil. The whole idea is to create an environment that supports a desired lawn more than it encourages the moss.
Other things (besides low pH) that encourage mosses are: shade, compacted soil, poorly draining soil (wet or dry) and poor fertility. These are all conditions antithetical to optimal grass growth. Anything you can do to alleviate these characteristics in your lawn will help your grass naturally "fight off" mosses.
Realize this is not necessarily a cure-all concept. Remember I said you are changing the natural environment to encourage certain flora and fauna, and discourage other flora and fauna. You could find that new "weeds" become problematic because they like and compete in the new environment that you create, better than before.
N.B. It doesn't matter what kind of hair cap moss you have. The general identity is good enough in this case, and recommendations would not change.
