if it was bright and white before suddenly going out, then the inverter is probably the problem.

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It was perfectly fine one day, and the next, it just didn't. My instincts thought it was the inverter, but until I get a chance to try a replacement, I can't be completely sure. Does the backlight bulb still get power if the inverter is the problem? Is there some way I could check with a voltmeter to make certain before I start ordering parts?

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If the inverter is bad, the typical failure mode is no or low voltage applied to the tube. When operational, a backlight gets on the order of 1000V AC from the inverter. A voltmeter is certainly a good way to determine whether the inverter is dead, but note the voltage, and take appropriate precautions. A functioning inverter can bite you very badly! Also note that the voltages may exceed the safe input range of many modern multimeters.

The suddenness with which the failure occurred is consistent with a failed inverter, but additional testing is the only way to be certain.