The authors review some basic facts from the spectral theory of Koopman operators defined by \(U_T(f)=f\circ T\) such that \(U_T:L^2(X,\mu)\rightarrow L^2(X,\mu)\). Then they describe the spectrum for rotations and characterize the ergodicity of the product of two transformations via their spectra. The authors prove that for every ergodic measure-preserving system \((X, T, \mu)\) there exists an irrational \(\alpha\) such that \((\mathbb{T}^1,R_{\alpha}, \lambda)\) and \((X, T, \mu)\) are disjoint. The main result of paper is given as follows: if \(X\) is a shift space with a safe symbol (in particular, if \(X\) is a hereditary shift) and \(t \geq 0\), then \(\mathcal{M}^{(t)} _{\sigma}(X)\) endowed with \(\overline{d}_{\mathcal{M}}\) is arcwise connected. Here \(\mathcal{M}^{(t)} _{\sigma}(X)\) is the set of ergodic measures on \(X\) with entropy less than or equal to \(t\), that is, \(\mathcal{M}^{(t)}_{\sigma}(X)=\{\mathcal{M}^{e}_{\sigma}(X):h(\mu)\leq t\}\) and \(h(\mu)\) is the metric entropy function. Furthermore the authors prove that if \(X\) is a shift space with a safe symbol (in particular, if \(X\) is a hereditary shift), then \(\{h(\mu) : \mu \in \mathcal{M}^{e}_{\sigma}(X)\} = [0, h_{\operatorname{top}}(X)]\) (possibly degenerate to a point). The authors present some notable examples of hereditary shifts to which the main result can be applied. The authors prove that for every Polish topological space \(P\) there exists a minimal shift space \(X\) such that \(\mathcal{M}^{e}_{\sigma}(X)\) with the weak\(^{*}\) topology and \(P\) are homeomorphic and there is a unique measure \(\mu \in \mathcal{M}^{e}_{\sigma}(X)\) with positive metric entropy.