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Piotr Hajlasz
Piotr Hajlasz
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On conditions for constructing integrable functionthe existence of $h\in L^1$ such that is non-zero almost anywhere$h>0$ a.e

So I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has one of the following proposition.

ThereIf $\mu$ is $\sigma$-finite, then there is a mapfunction $h \in \mathcal{L}^{1}(\mu)$$h \in L^{1}(\mu)$ such that $h > 0$ almost everywhere.
In the proof, ithe constructed an increasing setsequence of sets $A_{n}$ with limit being the whole space $\Omega$ and each element possessing finite measure, i.e. $\mu(A_{n}) < \infty$.

I am not sure whether this is only for $\mu$ being $\sigma$-finite or others, since in the following theorem he showed result similar to the Vitali convergence theorem but without the condition that restrict set measure.

On conditions for constructing integrable function that is non-zero almost anywhere

So I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has one of the following proposition.

There is a map $h \in \mathcal{L}^{1}(\mu)$ such that $h > 0$ almost everywhere.
In the proof, it constructed increasing set $A_{n}$ with limit being the whole space $\Omega$ and each element possessing finite measure, i.e. $\mu(A_{n}) < \infty$.

I am not sure whether this is only for $\mu$ being $\sigma$-finite or others, since in the following theorem he showed result similar to the Vitali convergence theorem but without the condition that restrict set measure.

On conditions for the existence of $h\in L^1$ such that $h>0$ a.e

I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition.

If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)$ such that $h > 0$ almost everywhere.
In the proof, he constructed an increasing sequence of sets $A_{n}$ with limit being the whole space $\Omega$ and each element possessing finite measure, i.e. $\mu(A_{n}) < \infty$.

I am not sure whether this is only for $\mu$ being $\sigma$-finite, since in the following theorem he showed result similar to the Vitali convergence theorem but without the condition that restrict set measure.

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Uriah
Uriah
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On conditions for constructing integrable function that is non-zero almost anywhere

So I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has one of the following proposition.

There is a map $h \in \mathcal{L}^{1}(\mu)$ such that $h > 0$ almost everywhere.
In the proof, it constructed increasing set $A_{n}$ with limit being the whole space $\Omega$ and each element possessing finite measure, i.e. $\mu(A_{n}) < \infty$.

I am not sure whether this is only for $\mu$ being $\sigma$-finite or others, since in the following theorem he showed result similar to the Vitali convergence theorem but without the condition that restrict set measure.