uniform approximation by a particular set of functions

Assume that the linear combinations of $\{f_i\}$ are not dense in $C[0,1]$. Then by Hahn - Banach theorem and Riesz - Markov - Kakutani theorem there exists a non-trivial Borel finite sign measure $\eta$ on $[0,1]$ such that $0=\int f_k(t)d\eta=\sum_{i=1}^n \int (\mu_i(t))^kd\eta(t)=\sum_{i=1}^n \int_{\mathbb{R}} x^k d\mu_i^*(\eta)$ (where $\mu_i^*(\eta)$ is the pullback of $\eta$). So the finite finitely supported Borel sign measure $\sum \mu_i^*(\eta)$ has zero moments, thus it is zero by the usual Weierstrass theorem on the segment. However if we denote by $a\in [0,1]$ the minimal element of the support of $\eta$, the point $\mu_1(a)$ belongs to the support of $\mu_1^*(\eta)$, but not to the supports of $\mu_i^*(\eta)$, $i>1$. A contradiction.