Consider $X$ as non-decreasing non-negative function. Consider $\mu$ and $\nu$ as two probability measures on $(\mathbb{R},\mathcal{B})$ for which we know $\mu([t, \infty)) \geqslant \nu([t, \infty)) \;\forall t \in \mathbb{R}$. How can I show that $\int Xd\mu \geqslant \int Xd\nu$?
I started from the definition of the integral of non-negative function but was not successful proving it. Then I thought since $X$ is monotone, maybe monotone convergence can help, but after 3 hours, I still don't know how to approach this. I appreciate if you could guide me.