A function $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ F : R 2 → R is sup-measurable if, for each (Lebesgue) measurable function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R , the Carathéodory superposition $$F_f:\mathbb {R}\rightarrow \mathbb {R}$$ F f : R → R given by $$F_f: x\mapsto F(x,f(x))$$ F f : x ↦ F ( x , f ( x ) ) is measurable. The existence of non-measurable sup-measurable functions is independent of ZFC. We prove, assuming CH, that the family of all non-measurable sup-measurable functions $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ F : R 2 → R (plus the zero function) contains a linear vector space of dimension $$2^\mathfrak {c}$$ 2 c . A function $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ F : R 2 → R is separately measurable if all its vertical and horizontal sections are measurable. In the second part of this note we show that the family of non-measurable separately measurable functions is $$2^\mathfrak {c}$$ 2 c -lineable.

中文翻译：

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两变量不可测函数族的线性化

函数 $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ F : R 2 → R 是可测的，如果对于每个（勒贝格）可测函数 $$f:\mathbb {R} \rightarrow \mathbb {R}$$ f : R → R ，Carathéodory 叠加 $$F_f:\mathbb {R}\rightarrow \mathbb {R}$$ F f : R → R 由 $$F_f: x\ 给出映射到 F(x,f(x))$$ F f : x ↦ F ( x , f ( x ) ) 是可测量的。不可测超可测函数的存在与 ZFC 无关。我们证明，假设 CH，所有不可测可测函数的族 $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ F : R 2 → R（加上零函数）包含维度为 $$2^\mathfrak {c}$$ 2 c 的线性向量空间。一个函数 $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ F : R 2 → R 如果它的所有垂直和水平部分都是可测量的，则它是单独可测量的。