## Sufficient and necessary conditions for integrable function

Define a function of $$y>0,$$ $$f(y)=\int_0^{1}x^{p-1}e^{-\frac{x}{y}}dx,$$ where $$p>0.$$ Denote $$I(a_1,a_2)=\iint_{0 Question: What is the sufficient and necessary conditions of $$(a_1,a_2)$$, satisfying $$I(a_1,a_2)<\infty$$?

## Definition of integrable representation of Kac-Moody algebra

I have seen several definitions of integrable representation $$V$$ of Kac-Moody algebra $$\mathfrak{g}$$ online. Which one is the standard one? Are they actually equivalent?

First one is $$e_i, f_i$$ acts nilpotently on $$V$$, where $$e_i, f_i$$ are the Chevalley basis of $$\mathfrak{g}$$.

Second one is for any root $$\alpha$$, restriction of $$V$$ to the $$sl_2$$ corresponding to $$\alpha$$ can be integrated to a $$SL_2$$ representation.

Third one is $$V$$ can be lifted to a representation of the (maximal) simply connected Kac-Moody group whose lie algebra is $$\mathfrak{g}$$.

The case I am interested most is untwisted affine Lie algebra. So feel free to restrict to this case if it helps.

## Sequence of integrable functions on [a,b] converges pointwise but not uniformly?

I’m currently taking Real Analysis and there was an example in the textbook showing that the sequence of integrable functions fn(x) converges pointwise. I’m a bit confused as to why that is and also why does it not converge uniformly? Here is the example in textbook!

Thank you!

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## Sequence of integrable functions on [a,b] converges pointwise but not uniformly?

I’m currently taking Real Analysis and there was an example in the textbook showing that the sequence of integrable functions fn(x) converges pointwise. I’m a bit confused as to why that is and also why does it not converge uniformly? Here is the example in textbook!

Thank you!

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## Projection of an invariant almost complex structure to a non integrable one

My apology in advance if my question is obvious or elementary

We identify elements of $$S^3$$ with their quaternion representation $$x_1+x_2 i +x_3 j +x_4 k$$. We consider two independent vector fields $$S_1(a)=ja$$ and $$S_2(a)=ka$$ on $$S^3$$. On the other hand $$P: S^3\to S^2$$ is a $$S^1$$-principal bundle with the obvious action of $$S^1$$ on $$S^3$$. Then the span of $$S_1, S_2$$ is the standard horizontal space associated to the standard connection of the principal bundle $$S^3 \to S^2$$. Then each horizontal space has an almost complex structure $$J$$. This is the standard structure associated to $$S_1, S_2$$ coordinate.

Is this structure invariant under the action of $$S^1$$? If yes, we can define a unique almost complex structure on $$S^2$$ which is $$P$$ related to the structure on total space. Now is this structure on $$S^2$$ integrable?

As a similar question, is there an example of a principal bundle $$P\to X,$$ such that $$P$$ is a real manifold and $$X$$ is a complex manifold and a connection admit an invariant almost complex structure which project to a non integrable structure?

## Proving that a function is Riemann-stieltjes integrable

Let $$g$$ a increasing function, and $$f$$ integrable with respect to $$g$$ in $$J=[a,b]$$ proof that $$|f|$$ is integrable with respect to $$g$$

By definition if $$f$$ is integrable with respect to $$g$$, for every partition $$P$$ of $$J$$ and for every $$\epsilon>0$$ exists $$I$$ in $$R$$ such that if $$Q$$ refine $$P$$ then $$|S(Q,f,g)-I|< \epsilon$$ $$|S(Q,f,g)-I|= \sum^{n}_{j=0}f(\lambda_{j})(g(x_{j}-x_{j-1})$$ then

$$\sum^{n}_{j=0}|f(\lambda_{j})|(g(x_{j}-x_{j-1}) \geq \sum^{n}_{j=0}f(\lambda_{j})(g(x_{j}-x_{j-1})$$

but i don’t know how to do

$$\epsilon \geq \sum^{n}_{j=0}|f(\lambda_{j})|(g(x_{j}-x_{j-1})$$

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## Prove that $f$ is not Lebesgue integrable

I need a hand with the following exercise:

Prove that $$f: (0,2) \to \mathbb{R}$$ given by

$$f(x) = \begin{cases} \frac{1}{x} & 0

Is not Lebesgue-integrable.

Here are my thought:

We can write $$f$$ as $$f = f^+ – f^-$$ where

$$f^+(x) = \begin{cases} f(x) & \text{if} \ f(x)>0 \ 0 &\text{otherwise}& \end{cases}$$

$$f^-(x) = \begin{cases} -f(x) & \text{if} \ f(x)\le0 \ 0 &\text{otherwise}& \end{cases}$$

And by definition $$f$$ is integrable if and only if $$f^+$$ and $$f^-$$ are both integrable, so I just need to prove that $$f^+$$ or $$f^-$$ is not Lebesgue integrable.

Now, in this particular case $$f^+(x) = \begin{cases} \frac{1}{x} & 0 and $$f^-(x) = \begin{cases} 0 & 0

So $$f = f^+-f^-$$ becomes $$f = \frac{1}{x}\chi_{(0,1]}-(-\frac{1}{x-1}\chi_{[1,2)})$$

thus if I proove that, says, $$\frac{1}{x}$$ is not Lebesgue integrable on $$(0,1]$$ the problem is solved. But how to prove that?

## Convexs functions are (Riemann) integrable

Let $$f: [0,1]\to\mathbb{R}$$ be a convex function. How to prove that $$f$$ is (Riemann) integrable?

Since $$f$$ is convex, one knows that $$f$$ is continuous on $$(0,1)$$. If $$f$$ were continuous on $$[0,1]$$ this would be enough to conclude.

May someone help me to conclude the proof?

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## Show Lebesgue Integrable and Compute the Two Iterated Integrals

(I am working on problems having to do with Fubini’s Theorem)

Given $$α ∈ (0,∞)$$, show that the function $$(x, y) \mapsto e^{−αxy}\cdot sin x$$ is Lebesgue integrable on $$(0,∞) × (1,∞)$$. Compute the two iterated integrals and use the result to compute

$$\int_0^{\infty} e^{\alpha x} \frac{sinx}{x}dx$$

How do I show the function is Lebesgue integrable? Usually I need to show that the Lebesgue integral is finite… but I am new to having two variables in these problems.

Now, for evaluating the integral. I have evaluated each of them below, then set them equal, as the iterated integrals should be equal. Is that correct?

dxdy

$$\int_1^{\infty} \int_0^{\infty} e^{-\alpha xy} \cdot sinx dxdy$$

$$I = \int_0^{\infty} e^{-\alpha xy} \cdot sinx dx$$

Let $$u = e^{-\alpha yx}, du = -\alpha ye^{-\alpha yx}, v = -cosx, dv = sinxdx$$.

$$I = -cosxe^{-\alpha yx}\rvert_0^{\infty} – \alpha y \int_0^{\infty}cosxe^{-\alpha yx}dx$$

Let $$u = e^{-\alpha yx}, du = -\alpha ye^{-\alpha yx}, v = sinx, dv = cosxdx$$.

$$I = (0-(-1)(1)) – \alpha y [e^{-\alpha yx}sinx\rvert_0^{\infty} + \alpha y \int_0^{\infty} e^{-\alpha xy} \cdot sinx dx]$$

$$I = 1 – \alpha y(0-0) – \alpha^2 y^2 I$$

$$I = \frac{1}{1+\alpha^2 y^2}$$

Now we have,

$$\int_1^{\infty} \frac{1}{1+\alpha^2 y^2}$$

Let $$u = \alpha x, du = \alpha dx$$.

$$= \frac{1}{\alpha} \int_{\alpha}^{\infty} \frac{1}{1+u^2} du = \frac{1}{\alpha} (arctan(\alpha x))\rvert_1^{\infty} = \frac{1}{\alpha} (\frac{\pi}{2} – arctan(\alpha))$$

dydx

$$\int_0^{\infty} \int_1^{\infty} e^{-\alpha xy} \cdot sinx dydx$$

$$=\int_0^{\infty} [\frac{sinx}{-\alpha x} \cdot e^{-\alpha xy}]\rvert_1^{\infty} dx = \int_0^{\infty} \frac{sinx}{-\alpha x} (0 – e^{-\alpha x}) dx = \frac{1}{\alpha} \int_0^{\infty} e^{-\alpha x} \cdot \frac{sinx}{x} dx$$

Then I set them equal to evaluate the integral the problem asks for.

$$\frac{1}{\alpha} (\frac{\pi}{2} – arctan(\alpha)) = \frac{1}{\alpha} \int_0^{\infty} e^{-\alpha x} \cdot \frac{sinx}{x} dx$$

$$\implies \frac{\pi}{2} – arctan(\alpha) = \int_0^{\infty} e^{-\alpha x} \cdot \frac{sinx}{x} dx$$

My issue is that in the problem is is $$\alpha x$$ not $$-\alpha x$$.

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## Prove bounded set implies $L^2$ integrable function?

On a complete and filtered probability space, $$(\Omega, \mathcal{F}, \mathcal{P})$$, I would like to show that a function $$f(t, \omega_1, \omega_2): \mathbb{R} \times \mathbb{R}^2 \to A \in \mathbb{R}$$ is $$L^2$$-integrable, i.e. $$\int_0^{\infty}E[f^2(s)]ds < \infty$$ if the set $$A$$ is bounded. I’m very new to analysis, so I could use some help.

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