Microwave is dead?

Most have an internal slow blow fuse.

just to be certain plug a lamp, hair dryer, anything into that outlet to be sure it's hot.

Microwaves have an internal safety (typically ceramic body) fuse. The fuse can blow for several reasons:
1. Age
2. Door misalignment, or gunk in the door switch. (The oven incorporates various safety mechanisms to cause the fuse to blow and cut off power if the door is open when the magnetron is active. Normally, a separate switch or part of the switch signals the control circuit to stop before the safety switch causes the fuse to blow, but if things are not aligned or something interferes with the sequencing, they might not activate in the proper order.)
3. Faults (mostly shorts) in the magnetron or high-voltage power supply circuit.
4. Probably a bunch of other (less common) things.

Be very careful if you open up the microwave to check the fuse. The high voltage supply is entirely capable of producing a lethal shock; it operates at a few kilovolts and a substantial fraction of an amp. Electrically speaking, a microwave is the most dangerous common household appliance to work on. The high voltage capacitor in the circuit can store a very nasty charge even when the oven is unplugged. There generally is a bleeder resistor/circuit to discharge the capacitor; however, especially as the oven is malfunctioning, it's wise to not rely on that working. There are some good safety guides available online on safely and effectively troubleshooting microwave oven problems.

Yes the do have an internal fuse. If it needs replacement the only thing "Special" about an RV Mirowave is venting.. Microwaves come in essentally several flavors when it comes to venting.. Some vent to the front, Rear, Top or one or the other side.

Depending on how the microwave is mounted you may need to "Duct" the vented high moisture air to a specific location.

To show you how "Special" RV microwaves are.. Mine is a "Dometic" it has a metal duct to bring the venting to the front. Open the door and you find out it's an LG (Lucky Goldstar) and like all goldstar products.. I was not lucky enough to get one that worked.. So I had to do a bit of engineering and work on it. IT now works perfectly (Well... I did go to school to study electronics and engineering).

Since the microwave is a kitchen appliance is there any chance that the outlet is a GFCI that has tripped? Mine was plugged in inside an upper cabinet.

Update my daughter and grandson came to join me last night and I had them bring my volt meter the outlet is dead. I now need to do some more checking I plan to check the pedestal to see if I am getting both legs coming in but it is wet outside and I have not felt like messing with it yet. My daughter also brought my shop microwave so I cooked my breakfast burrito with that this morning. I am a retired Lineman trouble shooter so this will get figured out. We are at the dunes riding our off road machines so I am not wanting to spend much time working on the MH until I get home.

The power to my microwave comes from the inverter which has a separate 20 circuit breaker on the inverter. Don't about yours but take a look

Hard to find breakers (pin style) on you inverter in a basement bay, usually on the ceiling. Use a flashlight to find all 3 of them. them

Yes it was the breaker on the inverter. I had bad internet coverage at the camp ground so I didn't get to check this until now and found the inverter breaker tripped when I got home now I know what to check if it happens again. Thanks to everyone who replied.