# Tagged Questions

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### Concerning a theorem relating Lebesgue integrability criteria on R^n

I have a question from Munkres' Analysis on Manifolds textbook, part of which i am having trouble with. Theorem: Let $S$ be a bounded set in $\mathbb R^n$; let $f\colon S\to\mathbb R$ be a bounded ...
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### Hardy-Littlewood maximal function weak type estimate

Show that if $f\in L^1(\mathbb{R}^d)$ and $E\subset \mathbb{R}^d$ has finite measure, then for any $0<q<1$, $$\int_E |f^{*}(x)|^q dx\leq C_q|E|^{1-q}||f||_{L^1(\mathbb{R}^d)}^{q}$$ where $C_q$ ...
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### Example of a countably infinite set that has a positive volume

Definition: The volume of a bounded set $A\subset\mathbb R^n$ whose characteristic function $1_A$ is integrable over $\mathbb R^n$ is $\int_A 1_A$. I'm looking for an example of a countably infinite ...
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### Understanding Stokes' theorem

Stokes' theorem( here I am only talking about the special $\mathbb{R}^3$ case) contains a line integral $\int_{\partial S} \langle f, \tau \rangle ds$. (Actually, I would be confident if somebody ...
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### counterexample for Dominated Convergence Theorem

The Dominated Convergence Theorem is as follows: What if the sequence $\left\{f_n \right\} \notin L^1$? Could someone provide a counterexample as to why the theorem wouldn't hold? Thanks!