Prove that, if (tau) is an initial time with (mathbb{E}_{mathbb{Q}}left(1 / alpha_{infty}^{tau} ight)

Prove that, if \(\tau\) is an initial time with \(\mathbb{E}_{\mathbb{Q}}\left(1 / \alpha_{\infty}^{\tau}\right)<\infty\), there exists a probability \(\widehat{\mathbb{Q}}\) equivalent to \(\mathbb{Q}\) under which \(\tau\) and \(\mathcal{F}_{\infty}\) are independent.