Assume that hypothesis ((mathcal{H})) holds under (mathbb{P}). Let [left.mathbb{Q} ight|_{mathcal{G}_{t}}=left.L_{t} mathbb{P} ight|_{mathcal{G}_{t}} ;left.quad mathbb{Q} ight|_{mathcal{F}_{t}}=left.widehat{L}_{t} mathbb{P} ight|_{mathcal{F}_{t}}]

Assume that hypothesis \((\mathcal{H})\) holds under \(\mathbb{P}\). Let

\[\left.\mathbb{Q}\right|_{\mathcal{G}_{t}}=\left.L_{t} \mathbb{P}\right|_{\mathcal{G}_{t}} ;\left.\quad \mathbb{Q}\right|_{\mathcal{F}_{t}}=\left.\widehat{L}_{t} \mathbb{P}\right|_{\mathcal{F}_{t}}\]

Prove that hypothesis \((\mathcal{H})\) holds under \(\mathbb{Q}\) if and only if:
\[forall X \geq 0, X \in \mathcal{F}_{\infty}, \quad \frac{\mathbb{E}\left(X L_{\infty} \mid \mathcal{G}_{t}\right)}{L_{t}}=\frac{\mathbb{E}\left(X \widehat{L}_{\infty} \mid \mathcal{F}_{t}\right)}{\widehat{L}_{t}}\]