How is the temperature of outer space measured?

I was just listening to the radio, and they were talking about how you would freeze to death if you were in outer space, because space is so cold.

I thought to myself, that this must be wrong, because heat needs to transfer in order to cool down, and in space there is nothing to transfer your heat to, so I think you would just float in space at body temperature.

Then I started to wonder, how exactly do you measure the temperature of space? Is it a mathematical calculation? or have we actually taken physical measurements?

I find it hard to imagine that you could just pull out a conventional thermometer while doing a space walk and see, so how exactly do you measure the temperature of space?

Do we actually have definitive first hand proof that outer space is cold? or just mathematical proof based on an estimate of matter per cubic meter or something?

Thanks.