Question

Lebesgue integrability of $f_n (x) := \begin{cases} \frac{\lvert\ln (n^7x^2)\rvert}{3+n^4x^2} & x \ne 0 \\ 1 & x=0\end{cases}$ on $\mathbb R$

score 43 · Answer 1 · 2023-02-12 00:09:55

Lebesgue integrability of $f_n (x) := \begin{cases} \frac{\lvert\ln (n^7x^2)\rvert}{3+n^4x^2} & x \ne 0 \\ 1 & x=0\end{cases}$ on $\mathbb R$

score 116 · Answer 2 · 2026-04-11 13:15:25

116

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score 61 · Answer 3 · 2023-02-12 13:59:23

61

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score 118 · Answer 4 · 2026-03-28 02:10:32

118

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score 61 · Answer 5 · 2023-02-13 08:58:44

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score 1.6k · Answer 6 · 2026-04-08 21:34:01

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156

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If the infinite sum of the norms $||f_n||$ converges then $\sum f_n$ converges to some $f \in \mathcal{L}_2(\Omega)$

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score 175 · Answer 8 · 2026-04-11 15:00:22

175

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32

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69

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score 22 · Answer 12 · 2023-02-17 17:01:14

22

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score 58 · Answer 13 · 2023-02-17 20:20:51

58

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score 87 · Answer 14 · 2023-02-18 18:26:45

87

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score 38 · Answer 15 · 2023-02-20 17:19:38

38

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List Question

Lebesgue integrability of $f_n (x) := \begin{cases} \frac{\lvert\ln (n^7x^2)\rvert}{3+n^4x^2} & x \ne 0 \\ 1 & x=0\end{cases}$ on $\mathbb R$

Easy Lebesgue integral of a non-horizontal line but by definition

Prove that $\lambda_{n}(tA)=t^n \lambda_{n}(A)$

When is the composition of Lipschitz and Lp still Lp?

Almost everywhere convergence and Lebesgue integrability of $\displaystyle\sum_{n = 1}^{+\infty} f_n (x)$

Why do we need Lebesgue Integration and what are the limits to Riemann Integrals?

If the infinite sum of the norms $||f_n||$ converges then $\sum f_n$ converges to some $f \in \mathcal{L}_2(\Omega)$

Questions about theorem 2.1 and lemma 2.2 in chapter 3 of Stein's Real Analysis

Is interchanging the integration and differentiation valid here?

Lebesgue integral on an open subset of $\mathbb R$

How to evaluate this integration

To show the inequality about $\mathcal L^1$ function.

Show that if $f\in L^1_{loc}(\mathbb{R})$ and $f(x)=\frac{1}{2r}\int_{x-r}^{x+r}f(t)dt$ for all $x\in\mathbb{R},r>0$ then $f\in C^{\infty}$

Product of a non negative decreasing function with a Lebesgue integrable function on $[0, 1]$

Is $1/f$ integrable when $f$ is finite on compact sets, absolutely continuous and $f \geq 1$?

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