# Questions tagged [lebesgue-measure]

For questions about the Lebesgue measure, a measure defined on the Borel or Lebesgue subsets of the real line or $\mathbb R^d$ for some integer $d$. Use it with (tag: measure-theory) tag and (if necessary) with (tag:lebesgue-integral).

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### Fatou's Lemma applied to simple functions

Show that the sequence of measurable functions $f_i: \mathbb{R} \rightarrow \mathbb{R}$ defined via \begin{align*}f_i(x)= \begin{array}{cc} \{ & \begin{array}{cc} -1 & i \leq ...
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### Given $f$ is a Lebesgue measurable function and $\int_0^1 x^{2n}f = 0 ~~~ \forall n$ , then show that $f = 0$ a.e.

Given $f$ is a Lebesgue measurable function and $\int_0^1 x^{2n}f\,d\mu = 0 \quad \forall n$, then show that $f = 0$ a.e. Of course, if it was given that $f \geq 0$ then this was pretty trivial. My ...
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### Prove that $\lim_{j\rightarrow\infty}\int_1^\infty\frac{f_j(x)}{x}dx=\int_1^\infty\frac{f(x)}{x}dx$ under these conditions…

Question: Let $\{f_j\}_{j\in\mathbb{N}}$ be a sequence of Lebesgue measurable functions satisfying $$\sup_{j\in\mathbb{N}}\int_1^\infty f_j^2(x)dx\leq1$$ such that $f_j\rightarrow f$ pointwise a.e. ...
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