This is an accessible introduction to the Ratner’s theorem on invariant measures. In fact, the author proves the following theorem: Let \(G\) be a Lie group, \(\Gamma\) a discrete subgroup of \(G\), which is isomorphic to \(\text{SL}(2,\mathbb{R})\). Then, any \(H\)-invariant and ergodic probability measure \(\mu\) on \(X=\Gamma\setminus G\) is homogeneous, i.e., there exists a closed connected subgroup \(L\) of \(G\) containing \(H\) such that \(\mu\) is \(L\)-invariant and some \(x_0\in X\) such that the \(L\)-orbit \(x_0L\) is closed and supports the measure \(\mu\). The author starts, as motivation, with the geodesic flows on the hyperbolic plane, unipotently generated subgroups, diagonalizable subgroups as well as some necessary ingredients of the proof such as the complete reducibility of \(\text{SL}(2,\mathbb{R})\) and the Mautner’s phenomenon for \(\text{SL}(2,\mathbb{R})\).