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This theorem has a generalization, which I think should be presented instead. For convergence in probability, it is as follows: if and are sequences of random variables, , and converges to zero in probability, then converges to zero in probability. This is given in the last pages of the cited paper by Mann & Wald.
The phase in the theorem that says "has the set of discontinuity points Dg such that Pr[X ∈ Dg] = 0 " would be clearer if the quantifier for the set Dg was made plainer. I think the condition Pr[X ∈ Dg] = 0" is intended to apply to each discontinuity point of the function. If so, the intended meaning is that each discontinuity point of the function satisfies the condition. However another interpretation of "has the set of discontinuity points" is that there exists a set of discontinuity points that satisfy the condition. By that interpretation, Dg need not contain all the discontinuity points.
![{\displaystyle \limsup _{n\to \infty }Pr[X_{n}\in D_{g}]=0}](/media/api/rest_v1/media/math/render/svg/4239480a851b081f817827293ccd8dee73ee12d5)
