I think overloading mathematical operators is very normal and common. Just like in programming though, one should be careful about what’s called “punning”.
i.e. it would be bad julia style to write
f(x::Number) = x + 1
f(x::String) = println("hello, world! $x")
Likewise, the function f , when used in a given context (i.e. namespace) should have a consistent meaning, even if the details of that meaning differ depending on the types of its input. As was mentioned above, addition and multiplication are heavily overloaded, but those overloads typically all preserve some core intuitions.
In the case you present here, I think the way you conflate the evaluation of a function at a point with it’s integral over a measure is maybe not ideal.