Yeah, I'd be a little leary of poking around the socket with meter leads, too easy to short out one of the leads. Might not get shocked, but the little arc weld you get will certainly get your attention.
Since it appears to be a magnetic ballast, it's likely the capacitor or igniter (possibly both) that is (are) the culprit(s). Either of these shouldn't be too difficult to source & replace. My first guess would be the capacitor. The transformer should be fairly bullet proof and that would be a more costly item to replace for sure.
Here's a link to a pdf that might save your bacon. The Pocket Guide to High Intensity Discharge Lamp Ballasts. Great info & a logical troubleshooting chart near the end. Includes a rather important caution when measuring voltage at the socket, make sure the ignitor is disconnected. Also talks about distance between ballast & ignitor being a possible issue. Apparently there is such a thing as a special ignitor for longer wire runs.
http://www.advance.philips.com/docum...HIDPcktGde.pdf