Not all inverters have electrolytic capacitors on them, but most do have some square capacitors that are not unmarked ceramic surface mounts but usually tantalum or electrolytic surface mounts.
Usually the scope probe does not even need connected, just place it near the inverter transformer and if it is running it will be clearly visible by a waveform that is not the usualy 60Hz noise waveform. I use this to confirm flybacks and sometimes horizontal coils to see if something is running or not. It does not tell me if it is running correctly though.
Start by checking the fuses, should be a main power in fuse and probably one on each output transformer and transistor feed. If any of the fuses are open, check for obvious shorted components. If no shorts are found, you can chance replacing the fuse, but there still may be a problem. Without a full schematic or lots of experience, it will be difficult to determine what caused the fuse to blow before replacing it.
If it seems to work with the new fuse, replace the lamps, they might be causing some excessive current draw due to age.