Let $\mathbb{T}$ denote the unit circle in $\mathbb{C}$, and let $\mu$ be a complex Borel measure on $\mathbb{T}$. Consider the space of square integrable functions $L^2\left(\mathbb{T},\mu\right)$.
Is the system $\left\{e^{int}\right\}_{n\in\mathbb{Z}}$ a complete system in $L^2\left(\mathbb{T},\mu\right)$?
Remark. Let $X$ be an Euclidean space. A subset $\left\{x_1,x_2,\dots\right\}$ is called a complete system, if $\overline{\text{Span}\left\{x_1,x_2,\dots\right\}}=X$.
My thought: by Weierstrass theorem, the system $\left\{e^{int}\right\}_{n\in\mathbb{Z}}$ is complete in $C\left(\mathbb{T}\right)$. Since $C\left(\mathbb{T}\right)$ is dense in $L^2\left(\mathbb{T},\mu\right)$, the result follows. Is this true?