Question on Chapman0Kolmogorov equation

Question

I am studying Chapman-Kolmogorov equation and here is the part in the chapter which I am confused about. I wonder by what theorem or condition that we can have equality of integral implies equality of the integrand. $$\int_{-\infty}^{\alpha}\int_{-\infty}^{\beta} P_{s,x}(t,dy)P_{X_s}(dx)=\int_{-\infty}^{\alpha}\int_{-\infty}^{\infty}\int_{-\infty}^{\beta} P_{u,z}(t,dy)P_{s,x}(u,dz)P_{X_s}(dx) $$ implies the equality $$P_{s,x}(t,A)=\int_{-\infty}^{\infty}P_{u,z}(t,A)P_{s,x}(u,dz)$$ for all $s<u<t,x\in\mathbb{R}$, and $A\in\mathcal{B}(\mathbb{R})$, the Borel field of $\mathbb{R}$.