I would look up the running voltage and put a fluke meter to it. If you have an electrician friend, or an electrical supply shop, you might be able to bring it to them and ask to use their meter.
Testing a ballast electrically is difficult because it has a high voltage spike (dangerous!) to light it and then a lower voltage to keep it lit. You would have to test both to know it's working.
Look for a thermal fuse they frequently go out. Open it and look for a capacitor with a bulging top. Check the cord (use a Fluke meter or swap for known good if you have one). Hope that helps and best of luck figuring it out.