query: Let $(X,S,\mu)$ is a measure space and $\mu(X)<\infty$. Define $d(f,g)=\int\frac{|f-g|}{1+|f-g|}d\mu$ is a metric on the space of measurable functions
positive_passages.docid: 52104
positive_passages.text: A pseudometric on the space of the measurable functions is complete
negative_passages.docid: 157426
negative_passages.text: Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$
negative_passages.docid: 53663
negative_passages.text: outer measure and measurable sets
negative_passages.docid: 143699
negative_passages.text: Bounded measurable function and integral with charcteristic function
negative_passages.docid: 56303
negative_passages.text: Why this definition for Lebesgue measurable functions?
negative_passages.docid: 151707
negative_passages.text: Fatou's lemma and measurable sets
negative_passages.docid: 121359
negative_passages.text: Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric.
negative_passages.docid: 83163
negative_passages.text: If $X$ is a connected metric space, then a locally constant function $f: X \to $ M, $M $ a metric space, is constant
negative_passages.docid: 109570
negative_passages.text: How to show that the set of points of continuity is a $G_{\delta}$
negative_passages.docid: 92638
negative_passages.text: Approximation of measurable function by simple functions
negative_passages.docid: 57462
negative_passages.text: Lebesgue Measure of the Graph of a Function
negative_passages.docid: 17067
negative_passages.text: To prove that $(\mathbb{P(N)},d)$ is a metric space
negative_passages.docid: 58827
negative_passages.text: Absolute value of a measurable function is measurable
negative_passages.docid: 21068
negative_passages.text: Is composition of measurable functions measurable?
negative_passages.docid: 88610
negative_passages.text: The set of convergence of a sequence of measurable functions is measurable
negative_passages.docid: 148759
negative_passages.text: The Radon-Nikodym derivative of a measure such that $|\int f'\,d\mu|\le \|f\|_{L^2}$ for $f\in C^1$
negative_passages.docid: 42084
negative_passages.text: Set of points of continuity are $G_{\delta}$