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If φis uniformly continuous on R to R, and if (fn) converges uniformly (respectively, almost uniformly, in measure) to f, then (φ∘fn. ) converges uniformly (respectively, almost uniformly, in measure) to φ∘f. Conversely, if φis not uniformly continuous, there exists a measure space and a sequence (fn) converging uniformly (and hence almost uniformly and in measure) to f but such that (φ∘fn) does not converge in measure (and hence not uniformly or almost uniformly) to φ∘f. | Numerade
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