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Hi, I'm Mike G and I'll be glad to assist you. Give me the count and colors of the wires in the box.
Is it switched controlled?
Is there more than one switch?
Do you remember which wires were connected to the old fixtures?
What point were you testing to. Wire to what, wires to what?
Retest from the single wire to the double connection, once with switch on and once with it off. Testing from a remote receptacle could and did give you that 240V reading. It did alert that the joint is the hot and the single wire is the neutral. They have the connections on the switch leg reversed. That happens alot in older home having work done by unqualified people.
OK, the wires are not reversed. Does the switch control the hot wire? If so, connect to those wires. Fan white to the 2 wire joint, fan black and blue to the single wire.
That's what is supposed to do.
You are confusing me. If testing the single black wire to a joint in the box, any joint. And getting 120V with the switch on and zero volts with the switch off. That is where you connect the fan. You have to use a 2 wire tester.