CONTINUITY OF MEASURE-DIMENSION MAPPINGS

The paper’s Theorem 2.8 (lower semicontinuity of dimU under setwise convergence, assuming countable stability) and Theorem 2.9 (upper semicontinuity of dimL under setwise convergence, no extra assumption) match exactly the task and are proven by the same subsequence/indicator-function arguments used by the candidate. The paper’s proofs hinge on: (i) countable stability to pass dimensions through a countable union in Theorem 2.8, and (ii) setwise convergence implying convergence of νn(A) for fixed measurable A via indicator functions in both theorems; the candidate’s solution mirrors these steps, choosing full-measure (resp. positive-measure) sets and applying setwise convergence to indicators to transfer nullity/positivity to the limit. Hence, both arguments are correct and essentially identical in structure to the official proofs in the paper. See the statements of Theorems 2.8 and 2.9 and their proofs in the paper for details ; the definitions of dimL, dimU and the setwise topology appear earlier in the paper .