query: Lebesgue integral of a bounded measurable function over a measurable subset of a measurable set of finite measure.
positive_passages.docid: 143699
positive_passages.text: Bounded measurable function and integral with charcteristic function
negative_passages.docid: 50730
negative_passages.text: Integral of bounded continuous function on $R$
negative_passages.docid: 113227
negative_passages.text: Measure spaces are proper subsets
negative_passages.docid: 66504
negative_passages.text: Proof that $f$ is Lebesgue-integrable and calculate its integral
negative_passages.docid: 97735
negative_passages.text: Continuous functions are differentiable on a measurable set?
negative_passages.docid: 60868
negative_passages.text: Interplay of Hausdorff metric and Lebesgue measure
negative_passages.docid: 69857
negative_passages.text: Is the set $A=\{x\in E|\lim_{n \to\infty} f_n(x)\ exists\ (finitely)\}$ measurable?
negative_passages.docid: 57206
negative_passages.text: Lebesgue measure on probability measure
negative_passages.docid: 69499
negative_passages.text: Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.
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: 98580
negative_passages.text: Lebesgue-measurable sets requiring the Axiom of Choice to construct
negative_passages.docid: 19632
negative_passages.text: Intuition behind the definition of a measurable set
negative_passages.docid: 69273
negative_passages.text: Is there a compact subset of the irrationals with positive Lebesgue measure?
negative_passages.docid: 148640
negative_passages.text: How to prove that if $\forall x \in (a,b)$ Lebesgue integral $\int_{(a,x)}fd\lambda=0$, then $f(x)=0$ $\lambda$-almost everywhere?
negative_passages.docid: 138055
negative_passages.text: If $A$ has a positive Lebesgue measure then there exist subsets which are not measurable
negative_passages.docid: 63309
negative_passages.text: How to find a measurable but not integrable function or a positive integrable function?
negative_passages.docid: 115219
negative_passages.text: If $E$ is Lebesgue measurable, show that there exists a closed set $F$ with $F \subset E$ and $m(E\setminus F)<\epsilon$