Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$ Here $\langle\cdot,\cdot\rangle$ denotes the inner product of functions with respect to the Lebesgue measure.

Is (a multiple of) Lebesgue measure the only Borel measure such that

- $T_\mu$ is defined for all $f\in C_c(\mathbb{R})$.
- $T_\mu(f)$ has exponential decay for all $f\in C_c(\mathbb{R})$.

Any reference is welcome.