Let \(H\) be a measurable map defined on \(\mathbb{R}^{+} \times \Omega \times \mathbb{R}\). Prove that, for any random time \(\tau\),

\[\int_{0}^{\tau} H\left(s, \omega, B_{s}\right) d s=\int_{-\infty}^{\infty} d x \int_{0}^{\tau} H(s, \omega, x) d_{s} L_{s}^{x}\]

where the notation \(d_{s} L_{s}^{x}\) makes precise that \(x\) is fixed and the measure \(d_{s} L_{s}^{x}\) is on \(\mathbb{R}^{+}, \mathcal{B}\left(\mathbb{R}^{+}\right)\).