Integrals of nonnegative functions are subadditive over covers and additive over disjoint unions

Let \((Y,\mcm,\mu)\) be a measure space, \(f\in\) $L^+$, and \(X,A,B\in\mcm\). If \(X\subseteq A\cup B\), then \[ \int_X f\dmu \leq \int_A f\dmu + \int_B f\dmu, \] with equality when \(X=A\cup B\) and \(A\cap B=\emptyset\).