On the output side of the trasformer, if you disconnect the lamp can you test the resistance across the leads with your multimeter set to ohms range? You should pick up the resistance of the lamp. Take the lamp out and confirm the resistance again.
If you get a relatively low reading for the first and an open circuit for the 2nd then all is well with the light and associated cabling hence I'd swop the transformer.
If you get a relatively low reading for the first and an open circuit for the 2nd then all is well with the light and associated cabling hence I'd swop the transformer.