I've read the solution of this exercize from this link Cohn Measure Theory exercise question 9 chapter 1.3 other direction
When I've tried to prove that $\lambda^* ( B_1 - B_0 ) = 0$ with the hint brought by one of the answer, I got $\mu^* ( \pi ( B ) \times \mathbb{R} ) = \mu^* ( \pi ( B ) \times \mathbb{R} \cap B ) + \mu^*( \pi ( B ) \times \mathbb{R} \cap \complement B )$, and so
$\lambda^* ( \pi( \pi ( B ) \times \mathbb{R} ) ) = \lambda^* ( \pi ( \pi ( B ) \times \mathbb{R} \cap B ) ) + \lambda^*( \pi ( \pi ( B ) \times \mathbb{R} \cap \complement B ) )$
We have $\lambda^* ( \pi( \pi ( B ) \times \mathbb{R} ) ) = \lambda^* ( \pi( B ) )$, $\lambda^* ( \pi ( \pi ( B ) \times \mathbb{R} \cap B ) ) = \lambda^* ( \pi ( B ) )$ and $\lambda^*( \pi ( \pi ( B ) \times \mathbb{R} \cap \complement B ) ) = \lambda^*(\pi(B) \cap \pi( \complement B ) ) = \lambda^*(\pi(B) - \complement \pi( \complement B ) ) = \lambda^*(B_1 - B_0)$
As a result, $\lambda^* ( \pi( B ) ) = \lambda^*(\pi(B)) + \lambda^*(B_1 - B_0)$
My problem is I don't know how to deal with the case when $\lambda^*(\pi( B )) = +\infty$ as I can't simply cancel out the $\lambda^* ( \pi( B ) )$.