query: How to prove $\mu(X \cup Y) \geq \mu(X) +\mu(Y) $ for this measure.
positive_passages.docid: 19773
positive_passages.text: Is finite additivity a property of the Lebesegue outer measure?
negative_passages.docid: 108560
negative_passages.text: Proof of a Gaussian Integral property
negative_passages.docid: 130066
negative_passages.text: If $\ \sum_{k=1}^n m(E_n) > n-1,$ then prove that $\bigcap_{k=1}^n E_k$ has positive measure.
negative_passages.docid: 123806
negative_passages.text: How do I show that the integral of $e^{inx}$ over a set of measure $1$ is nonzero for some nonzero $n$?
negative_passages.docid: 79165
negative_passages.text: How to show that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
negative_passages.docid: 27564
negative_passages.text: Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
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: 8007
negative_passages.text: Proof of Young's inequality
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: 27705
negative_passages.text: Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
negative_passages.docid: 116387
negative_passages.text: How to prove that $\Gamma(x) \leq x^x$ for $x \geq 1$
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 29849
negative_passages.text: How can I show that $\sup(AB)\geq\sup A\sup B$ for $A,B\subset\mathbb{R}$ where $A\cup B$ is positive and bounded?
negative_passages.docid: 138959
negative_passages.text: On the equality case of the Hölder and Minkowski inequalities
negative_passages.docid: 63124
negative_passages.text: Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
negative_passages.docid: 33617
negative_passages.text: Prove the inequality involving multiple integrals
negative_passages.docid: 37015
negative_passages.text: Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$