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There is a function F : [c]<ω → {0,1} such that if A ⊆ [c]<ω is uncountable, then {F(a ∪ b) : a, b ∈ A, a ≠ b} = {0, 1}. A corollary is that there is a function f : ℝ → {0, 1} such that if A ⊆ ℝ is uncountable, 2 ≤ k < ω, then both 0 and 1 occur as the value of f at the sum of k distinct elements of A. This was originally proved by Hindman, Leader, and Strauss under CH, and they asked if it holds in general. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two inequalities for generalized rational functions of one variable in Lp spaces when a partition of the domain with a suitable number of measurable subsets is considered. [ABSTRACT FROM AUTHOR]

We prove the existence of three subspaces of C[0, 1], each homeomorphic to C[0, 1]; the first consists only of infinitely many times differentiable functions, the second consists only of singular functions of bounded variation, and the third consists only of nowhere differentiable functions. [ABSTRACT FROM AUTHOR]

In this article we investigate the smallest (in the sense of inclusion) δ-field of subsets of R in which all functions of some families of functions from R to R satisfying some approximate quasicontinuity conditions introduced in [2] are measurable. [ABSTRACT FROM AUTHOR]