Laplace Transform of non integrable function

Question

Consider $$f(t)=\begin{cases}1 &x\text{ is rational}\\0&x\text{ is irrational}\end{cases}$$What is the Laplace transform of $f(t)$? I expect it to be $0$ but I am not able to prove it. Does such a function $f(t)$ qualify as a Null function?

Please help.

Thanks in advance.