Okay, hadn't given this one much thought. Here's one proof for continous functions under slightly different assumptions; and although it is an improvement from the case I'm still interested in the general case.

Let open and be such that for all and such that we have .

Let be given by where is such that . Define . It is a standard result that if then where .

With this in mind, let be continous then:

This proves that is harmonic in . Now the issue here, if we wanted to generalize this procedure to only integrable (or locally so), is that has measure zero so our change into "polar coordinates" can not be applied as we did here.

Any comments or suggestions are appreciated.