# Tagged Questions

0answers
64 views

### Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f$ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$

Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever \$\mu(E) < ...
2answers
85 views

### Solve limits in Lebesgue integral

Solve the limits of below: (1) $\lim\limits_{n \to \infty} \int_0^n (1+\frac{x}{n})^n e^{-2x}dx$. (2) $\lim\limits_{n \to \infty} \int_0^n (1-\frac{x}{n})^n e^{\frac{x}{2}}dx$. (3) ...
1answer
47 views

### The Lebesgue Theory basic Application , get stuck

Ok, I am working on a very easy question but I get stuck when I trying to justify my answer. I know that, in order to use Lebesgue's dominated Convergence Theorem, there are two conditions that we ...
0answers
70 views

### $\lim_{n \to \infty} \int^n_{-n}fdm=\int fdm$

Let $f:\mathbb{R} \to \mathbb{R}$ such that $f$ is integrable over $[-n,n]$ for every $n \in \mathbb{R}$ and assume that $$\lim_{n \to \infty} \int^n_{-n}fdm < \infty.$$ Proposition: $f$ is ...
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 21, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9191791415214539, "perplexity_flag": "head"}