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Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I} E(X_i) <\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{n}{n}]} \cdot n$$X_n=1_{[0,\frac{1}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I} E(X_i) <\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{n}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I} E(X_i) <\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{1}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).

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Sofia
Sofia
  • 11
  • 4

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I}<\infty$$sup_{i \in I} E(X_i) <\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{n}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I}<\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{n}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I} E(X_i) <\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{n}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).

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Sofia
Sofia
  • 11
  • 4

Looking for a family of random variables such that only the second clause is fulfilled

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I}<\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{n}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).