Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics
Linux Systems
Linux Systems
Power Users
Power Users
Tabletop RPGs
Tabletop RPGs
Notifications
Mark all as read
Q&A

$\int_{E_n} |g|^q = \left| \int_E \chi_{E_n}\cdot \text{sgn}(g)\cdot g \cdot |g|^{q-1}\cdot |g| \right|$

+4
−0

I am trying to understand why the following equation is true. Here $E$ is a measurable set and all functions are defined and measurable on it. $1<p,q,<\infty$ such that $\frac 1 p+\frac 1 q=1$ and $g\in L^q(E)$. $E_n= \{ x \in E:|g|\le n \}$. And there exists a number $M$ such that for every $f\in L^p(E)$, we have $\left|\int_Efg\right|\le M ||f||_p$.

$$\int_{E_n} |g|^q = \left|\int_E \chi_{E_n}\cdot \text{sgn}(g)\cdot g \cdot |g|^{q-1}\cdot |g|\right|$$

My main problem with this is that it seems like the powers are off by one. I would think that on $E_n$,

$$|g|^q = \chi_{E_n}\cdot |g|\cdot |g|^{q-1} = \chi_{E_n}\cdot\text{sgn}(g)\cdot g \cdot |g|^{q-1}$$

But even if we correct the extra factor of $|g|$ then I don't understand how we can write this as an equality. If we're using the "integral triangle inequalty" or whatever $\left|\int_Ef\right|\le\int_E|f|$ is called, then shouldn't the equality actually be an inequality? Is there some reason why in this particular setting we can actually have equality?

Why does this post require moderator attention?
You might want to add some details to your flag.
Why should this post be closed?

0 comment threads

0 answers

Sign up to answer this question »

This community is part of the Codidact network. We have other communities too — take a look!

You can also join us in chat!

Want to advertise this community? Use our templates!

Like what we're doing? Support us! Donate