Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space, i.e. a measure space such that \(\mathbb{P}(\Omega)=1\). Show that for a measurable function \(T: \Omega ightarrow[0, \infty)\) and every \(\lambda>0\) the following formula holds:

\[
\int e^{-\lambda T} d \mathbb{P}=1-\lambda \int_{0}^{\infty} e^{-\lambda s} \mathbb{P}(T \geqslant s) d s
\]

 What happens if we also allow negative values of \(\lambda\) ?