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Golshani, Mohammad. "Adding many random reals may add many Cohen reals". arxiv.org.Let $U \supseteq S$ be an open set with $\mu(U) < 2\cdot \mu(S).$
By continuity of addition, we can find an open set $V$ containing the zero function $0$ such that $V+S \subseteq U.$
We show that $V \subseteq S - S$. Thus suppose $x \in V.$ Then $(x+S) \cap S \neq \emptyset,$ as otherwise we will have
$(x+S) \cup S \subseteq U,$ and hence $\mu(U) \geq 2\cdot \mu(S),$ which is in contradiction with our choice of $U$.
Thus let $y_1, y_2 \in S$ be such that $x+ y_1 = y_2.$ Then $x= y_2 - y_1 \in S - S$ as required.
我理解后面的部分：$S$ 和 $x+S$ 的交集必须非空，否则不能将它们放入$U$，$\mu(U)<\mu(x+S)+\mu( S)$.
我不确定“By continuity of addition”部分的证明:
为什么可以找到一个包含 $0$ 的开集 $V$ 使得 $V+S \subseteq U$？ |
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