Mark GetHandler as const since it doesn't change anything.
Actually initialize the ServeMux after we had to move it to separate code.
Note that we already sent out the HTTP response header so we don't do it
twice. This is not mutex protected because we don't really expect
different threads to write to the same responsewriter.
Pass the request through to the handler.
Add a schema to the request so it can be represented as an URL in a more
appropriate fashion.