query: Prove the following property of a measure: $\mu ((0,x]) = cx.$
positive_passages.docid: 157322
positive_passages.text: Translation invariant measures on $\mathbb R$.
negative_passages.docid: 121504
negative_passages.text: Supremum of a product measurable function...
negative_passages.docid: 98823
negative_passages.text: Let $g:\mathbb{R}\to\mathbb{R}$ be a measurable function such that $g(x+y) =g(x)+g(y).$ Then $g(x) = g(1)x$ .
negative_passages.docid: 4064
negative_passages.text: An equality involving the mean value property
negative_passages.docid: 158494
negative_passages.text: Properties of special rectangle (measure)
negative_passages.docid: 65334
negative_passages.text: Show that a certain set is measurable
negative_passages.docid: 96286
negative_passages.text: Proving that sum of two measurable functions is measurable.
negative_passages.docid: 131793
negative_passages.text: Zero integral of measurable $f$ on every interval implies $f=0$?
negative_passages.docid: 67292
negative_passages.text: Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$
negative_passages.docid: 152310
negative_passages.text: If $E$ has Lebesgue measure $0$, must there exist a translate such that $E\cap E+x=\varnothing$?
negative_passages.docid: 80362
negative_passages.text: Simpler proof - Non atomic measures
negative_passages.docid: 118353
negative_passages.text: For set of positive measure $E$, $\alpha \in (0, 1)$, there is interval $I$ such that $m(E \cap I) > \alpha \, m(I)$
negative_passages.docid: 16743
negative_passages.text: Graph of real continuous function has measure zero
negative_passages.docid: 74583
negative_passages.text: Prove that $f'(0)=L$.
negative_passages.docid: 95260
negative_passages.text: Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.
negative_passages.docid: 101267
negative_passages.text: Show: $\int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)>t\right\})\, dt$
negative_passages.docid: 130537
negative_passages.text: Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$