Let (T,B,μ) be a measure and let f:ˉR×T→ˉR be a function. Then we say that f is an increasing μ-partition of unity if f(x,t) is increasing in x. measurable in t and ∫Tf(x,t)μ(dt)=x for all x∈ˉR. Increasing partitions of unity have a variety of applications which will be explored in the paper. For instance, applications include the Fubuni-Tonelli theorem for upper and lower integrals and Fubuni-integrals, measurability or upper (lower) semicontinuity of integral transforms, and construction of functions with a prescribed integral transform and satisfying a given set of (in)equalities.