The induced measures generated by analytic functions are under investigation in the present paper. Given a function \(\phi\) from the Hardy algebra \(H^\infty\), the induced measure \(\mu_{\phi}(E)\) is defined by the formula \[ \mu_{\phi}(E)=\frac 1{2\pi} |\{e^{i\theta}: \phi(e^{i\theta})\in E\}| \] for any measurable set \(E\subset \mathbb C\), where \(|\cdot|\) is the Lebesgue measure on the unit circle. The connection between induced measures, the Nevanlinna counting functions and the composition operators are discussed. The main problem under consideration is to characterize measures \(\mu\) compactly supported on the complex plane which can come up as induced measures. The author presents three necessary conditions and conjectures that they are also sufficient for \(\mu=\mu_{\phi}\) with some \(\phi\in H^\infty\). The result (theorem 1) proved in the paper provides a strong support for the author’s conjecture. It is clear that the normalized Lebesgue measure on any circle \(\{|z|=r\}\) is the induced measure \(\mu_{\phi}\) for \(\phi\) being a constant multiple of an inner function. The author shows (theorem 2) that there is \(\phi\in H^\infty\) such that \(\mu_{\phi}\) is one-half of normalized Lebesgue measure on \(\{|z|=1\}\) and one-half of normalized Lebesgue measure on \(\{|z|=2\}\). The latter result gives a negative answer on a question by W. Rudin.