Integrable majorant for a family

An integrable majorant for a parameterized family $f_\lambda$ is a nonnegative $g\in L^1(X,\mu)$ such that $|f_\lambda(x)|\le g(x)$ almost everywhere for every relevant parameter $\lambda$ . To use a pointwise limit argument, specify the exceptional null sets and, when necessary, a common null set for the parameter neighborhood.

Uniform control for limits

For a sequence converging almost everywhere, this hypothesis permits dominated convergence. Having a bound $\int|f_\lambda|\le C$ alone is weaker: the functions $f_n=n\mathbf1_{(0,1/n)}$ converge to zero almost everywhere on $(0,1)$ but their integrals stay one.

Parameter derivatives

For differentiation under an integral, the relevant majorant usually bounds the parameter derivative throughout a neighborhood, so that it also bounds the difference quotients by the mean value theorem.