query: Show that $\|f\|_{p}\leq \|f\|_{\infty}$ for any $0<p<\infty$ in a general measure space $X$ with $\mu(X)=1$.
positive_passages.docid: 151558
positive_passages.text: Integral over null set is zero
negative_passages.docid: 98098
negative_passages.text: Show that $\mu_{F}$ is a measure on $(\mathbb{R},\mathcal{S})$.
negative_passages.docid: 83825
negative_passages.text: $f^{-1}([-\infty,r))$ measurable for all rational numbers $r$
negative_passages.docid: 83959
negative_passages.text: If $f \in L_p \cap L_q$, $1 \leq p < r < q < \infty$ then $f \in L_r$.
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 $m(E\setminus F)<\epsilon$
negative_passages.docid: 74501
negative_passages.text: $\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.
negative_passages.docid: 84263
negative_passages.text: A Lebesgue measure question
negative_passages.docid: 9159
negative_passages.text: $L^p$-norm of a non-negative measurable function
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: 64486
negative_passages.text: Show that for non-negative measurable functions $f,g$ with $fg \geq 1$ the inequality $(\int f^p)(\int g^p) \geq 1$ holds.
negative_passages.docid: 23436
negative_passages.text: $f$ Holder continuous with Holder exponent $p>1\implies f \text{ is constant}$
negative_passages.docid: 93806
negative_passages.text: $\|f\|_{L^{p}}=\sup\left\{\int_{{\bf R}^{n}}|f(x)g(x)|~dx:\|g\|_{L^{q}}\le1\right\}$
negative_passages.docid: 20764
negative_passages.text: Is it true that $\lim\limits_{x\to\infty}{x·P[X>x]}=0$?
negative_passages.docid: 148582
negative_passages.text: Proof: $\|X\|_2\leq \sqrt{n}\|X\|_\infty$
negative_passages.docid: 155974
negative_passages.text: Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$
negative_passages.docid: 112323
negative_passages.text: Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$
negative_passages.docid: 63889
negative_passages.text: Prove that if $u_n\rightarrow \infty$ then $\frac{1}{u_n} \rightarrow 0$