AC Compressor was not turning off
If a repair fixes a problem, then in general one does not look for more things to do. Wait before you do anything more.

In my experience the 240 V power line goes directly to the condensing unit outside. In the US there is usually no 240 V line to the air handler inside. In the US most residential air handlers use 120 V on an entirely different circuit from the 240 V line to the condensing unit outside. 

There is a "relay" (called the contactor unit) inside the condensing unit which switches off and on the 240 V high current power to the compressor and fan. Failure of the contactor unit is common, but at present you have not given any reason to suspect yours should be replaced. 

EDIT

But in Europe there is only one voltage 230 V to 240 V so the air handler would be at 240 V. I assume that the air handler and the condensing unit would still be on different circuits, but I really don't know.