# Tagged Questions

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### Adding integrals with different domains

Suppose I have two integrals $$\int_{\Omega_1} f \, \, d \eta$$ and $$\int_{\Omega_2} g \, \, d \eta$$ how would I define the sum of these two integrals? Is it possible? I want something of the ...
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### Can this exercise be solved by DCT, I was only able to use MCT.

How would you solve this exercise? You don't need to give me the details, just the general idea. Let f be a Lebesgue integrable function. Show that $\int f(x+a) d\lambda=f(x) d\lambda$ and ...
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### lebesgue's integral and theorems [closed]

I have got an exercise from Lebesgue's integral: $$\lim_{n \to \infty} \int_{A} x^n y^{2n} \, dl_2 (x,y), \ A=\{ (x,y) \in \mathbb{R}^2 \mid 4x^2+y^2 \le 1 \}$$ I do not really understand Lebegues's ...
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### Proof monotone convergence theorem, why do they use this lim sup?

I have a question about the proof of the MCT. First they use a lemma, this is ok, but I'll show it for completeness: Now comes the proof. But I am wondering, why do they use a lim sup here?, why ...
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### Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.

Let $f(x)$ be a non-decreasing function on $[0, 1].$ You may assume that $f$ is differentiable almost everywhere. Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$. I am having a hard time with this ...
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### (a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ R$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that is also ...
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### Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Let $I = [a, b], E \subset I, m(E) = 0$ (but $E$ not empty). Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$. I am ...
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### Proof of FTC, continuity part, for Lebesgue integrable functions

The part of the FTC I am interested in says: If $f$ is a Lebesgue-integrable function on $[a,b]$, then $F(x)=\int_a^xf(t)\,dt$ is continuous. This is usually considered a lemma or something for ...
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### Does it make sense to talk about the integral of measurable functions that are not absolutely integrable?

Suppose $f$ is a real-valued (possibly infinite-valued) function on some measure space $(X, \Sigma, \mu)$, and suppose that it is measurable. Note that $f$ is not necessarily nonnegative. Does it ...
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### Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty$ and $\int f^{-} d\mu < \infty$ ...
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