Considering that we don't know the value of pi (not that we could write it out nor read it), I'm not sure your definition of "measure" is the same as mine or most people's.
I think it's reasonable to say we can't truly measure pi, though.
And you can neither know nor measure a random real.
Sure, but how would you compare those against a measurement?
> We know pi in the sense of "a unique real number satisfying many useful properties".
We know it a lot better than that. We have efficient programs that output the numerical value of pi for as many digits as you want.
There's a bunch of real numbers we can identify that are far harder to make use of or approximate, and don't have easy exact description of their value.