query: If f is a monotone function which defined on interval, then f is measurable
positive_passages.docid: 147546
positive_passages.text: $f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable
negative_passages.docid: 126698
negative_passages.text: measurability of a function -equivalent conditions
negative_passages.docid: 152563
negative_passages.text: Set of discontinuity of monotone function is countable
negative_passages.docid: 7681
negative_passages.text: How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?
negative_passages.docid: 60032
negative_passages.text: If the measure of union = sum of outer measures, then the sets are measurable
negative_passages.docid: 19632
negative_passages.text: Intuition behind the definition of a measurable set
negative_passages.docid: 109301
negative_passages.text: Lebesgue Integral of Non-Measurable Function
negative_passages.docid: 122585
negative_passages.text: Show that a function almost everywhere continuous is measurable
negative_passages.docid: 98098
negative_passages.text: Show that $\mu_{F}$ is a measure on $(\mathbb{R},\mathcal{S})$.
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: 137257
negative_passages.text: The Supremum and Infimum of a sequence of measurable functions is measurable
negative_passages.docid: 151707
negative_passages.text: Fatou's lemma and measurable sets
negative_passages.docid: 152877
negative_passages.text: Is a monotone differentiable function continuously differentiable?
negative_passages.docid: 14578
negative_passages.text: Proving that an additive function $f$ is continuous if it is continuous at a single point
negative_passages.docid: 82236
negative_passages.text: Extension of the Lebesgue measurable sets
negative_passages.docid: 125910
negative_passages.text: Continuity of Integration for a measurable function
negative_passages.docid: 148328
negative_passages.text: Example of a function $f$ on $X$ to $\mathbb{R}$ which is not $X$-measurable but $|f|$, and $f^2$ are measurable.