Not normal.  Can you charge it with a charger?  In any case that is
better than using the converter in the trailer.

However, there should be 12 volts from the converter to the battery as
well as 12 volts from the tow vehicle.

When you go to replace get a true deep-discharge battery instead of marine.

For information about batteries there are several sites such as:

http://www.batteryfaq.org

Hi,
If the battery is dead, no voltage, when you plug the RV into 110V then
it might not charge anyway.  Batteries will go bad from not being used,
they need to be kept charged even when not being used for months.  A
completely dead battery sometimes will not take a charge, or take a very
small charge then gradually build up and start taking a larger charge.

There should be a converter/charger somewhere.  You should be able to
check the output of the charger section of the converter by using a volt
meter.  When the charger is working you should see over 13.4 Volts or
more at the output of the charger and on the battery.   If you don't see
that high of a voltage, but only the battery voltage 12.7 volts or less
the charger is not working.  The 110V shore power should go to a circuit
breaker or fuse then to the converter/charger.  Check to see that the
circuit breaker or fuse for the converter is not tripped or blown.  Some
converter/chargers also have fuses built in.  You would need the manual
for the converter to determine if it has fuses.

Hope this helps some.  Take care and Happy Campin...