query: Prove $A = \{x ∈ X : f(x) ≥ a\}$ has finite measure
positive_passages.docid: 142812
positive_passages.text: Prove that $f\in L^1(A)\Leftrightarrow \sum_{n}^{\infty}m(\{ x\in A : f(x)\geq n \}) < \infty$
negative_passages.docid: 23765
negative_passages.text: Prove that lebesgue integrable equal lebesgue measure
negative_passages.docid: 65334
negative_passages.text: Show that a certain set is measurable
negative_passages.docid: 53627
negative_passages.text: Proving continuity of $f$
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: 45792
negative_passages.text: Rigorous proof that surjectivity implies injectivity for finite sets
negative_passages.docid: 27615
negative_passages.text: Prove Borel Measurable Set A with the following property has measure 0.
negative_passages.docid: 26369
negative_passages.text: Proving seemingly obvious equations with measure theory
negative_passages.docid: 88418
negative_passages.text: Proof of a theorem saying that we can make a measurable function continuous by altering it by a set of arbitarily small measure
negative_passages.docid: 121771
negative_passages.text: Proof $f(x)\equiv 0$
negative_passages.docid: 91947
negative_passages.text: Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
negative_passages.docid: 98897
negative_passages.text: Proving that the set of $\nu$ measurable sets $M\subseteq P(X)$ is an algebra
negative_passages.docid: 147021
negative_passages.text: Proving a function is bounded
negative_passages.docid: 21879
negative_passages.text: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A
negative_passages.docid: 47187
negative_passages.text: Proving the measure of an increasing sequence of measurable sets is the limit of the measures
negative_passages.docid: 19289
negative_passages.text: complement of a set and its measure
negative_passages.docid: 69499
negative_passages.text: Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.