Tagged Questions

For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$\int f d\mu$$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...
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Equivalent conditions for weak $L^p$ spaces for $p\leq 1$

I have difficulty doing the following exercise from Tao's real analysis book: Let $X$ be $\sigma$-finite measure space and $0<p\leq 1$. Then show that the following are equivalent: $f$ is in ...
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Show that there exists a continuous function $f$ such that $\int |\chi_A-f| d\lambda\lt \epsilon$

Let $\lambda=l^*$ denote Lebesgue measure on $\Bbb R$, and let $A$ be a Lebesgue measurable set with $\lambda(A)\lt +\infty$. Show that if $\epsilon \gt0$, there exists an open set which is the union ...
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Existence of a locally essentially unbounded integrable function

Does there exist an integrable function $f\colon [0,1]\to \mathbb{R}_+$ such that for every $0\leq a < b\leq 1$ we have $\| \chi_{(a,b)} f\|_\infty = + \infty$?
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Decreasing sequence of non-negative Lebesgue measurable functions and MCT

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem: Suppose that $f$ and $f_n$ are nonnegative measurable ...
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Verification of a proof in Measure Theory

Let $m$ be the Lebesgue measure on $\Bbb R$ and $f:\Bbb R\to [0,\infty)$ be a Lebesgue integrable function. Show that $\exists$ a measurable set $E\subset [0,\infty)$ such that $m(E)\neq m(f^{-1}(E)$ ...
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Two questions on Lebesgue integration and application of reverse triangle inequality

I'm learning about measure theory (specifically Lebesgue integration) and need help with the following problem: Let $f_n, f \in L^1$ and $\int_{\mathbb{R}}\left|f_n-f\right| \rightarrow0$. Prove ...
$\int_{\bigcup_{n=1}^{\infty}E_n}f=\sum_{n=1}^{\infty}\int_{E_n}f$ given $f$ positive and measurable
I'm learning about measure theory (specifically Lebesgue intregation) and need help with the following problem: Let $f:\mathbb{R}\rightarrow[0,+\infty)$ be measurable and let $\{E_n\}$ be a ...
Find the $\lim\limits_{c \to \infty} \int_{-\infty}^{+\infty} |g(x) - g(x+c)|dx$
Find the $$\lim_{c \to \infty} \int_{-\infty}^{\infty} |g(x) - g(x+c)|dx$$ where $g$ is integrable. I know already that if $g$ is integrable, than the integral of $g(x)$ and the integral of $g(x+c)$...