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Some questions on The applicability of the Dominated Convergence theorem on the real line

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now itsit is clear that $f_n\rightarrow 0$, but in the text I am using it says we can't apply the Dominated Convergence theorem as there is no function to dominate $f_n$, but 

I have trouble seeing that. To me the $f_n$'s are dominated by the constant function $1$. So I think can apply the Dominated Convergence theorem in some cases which depends on the choice measure where the constant functions are integrable. So if we are using the Lebesgue measure (on $\mathbb{R}$) then we can't use the Dominated Convergence theorem  (and the text is right), but if are using a finite measure then will it be valid to use the Dominated Convergence theorem and we will then have $\lim\limits_n\int f_n\rightarrow0$. Also if our space was just a bounded interval and we have the Lebesgue measure (since it is now a finite measure on the internal), then it will will it be fine to use the Dominated Convergence theorem on the $f_n$'s ? Is this right or wrong please let me know if am wrong and correct me. Thanks.?

Some questions on the Dominated Convergence theorem

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now its is clear that $f_n\rightarrow 0$, but in the text I am using it says we can't apply the Dominated Convergence theorem as there is no function to dominate $f_n$, but I have trouble seeing that. To me the $f_n$'s are dominated by the constant function $1$. So I think can apply the Dominated Convergence theorem in some cases which depends on the choice measure where the constant functions are integrable. So if we are using the Lebesgue measure (on $\mathbb{R}$) then we can't use the Dominated Convergence theorem(and the text is right), but if are using a finite measure then will it be valid to use the Dominated Convergence theorem and we will then have $\lim\limits_n\int f_n\rightarrow0$. Also if our space was just a bounded interval and we have the Lebesgue measure (since it is now a finite measure on the internal), then it will it be fine to use the Dominated Convergence theorem on the $f_n$'s ? Is this right or wrong please let me know if am wrong and correct me. Thanks.

The applicability of the Dominated Convergence theorem on the real line

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now it is clear that $f_n\rightarrow 0$, but in the text I am using it says we can't apply the Dominated Convergence theorem as there is no function to dominate $f_n$ 

I have trouble seeing that. To me the $f_n$'s are dominated by the constant function $1$. So I think can apply the Dominated Convergence theorem in some cases which depends on the choice measure where the constant functions are integrable. So if we are using the Lebesgue measure (on $\mathbb{R}$) then we can't use the Dominated Convergence theorem  (and the text is right), but if are using a finite measure then will it be valid to use the Dominated Convergence theorem and we will then have $\lim\limits_n\int f_n\rightarrow0$. Also if our space was just a bounded interval and we have the Lebesgue measure (since it is now a finite measure on the internal), then will it be fine to use the Dominated Convergence theorem on the $f_n$'s ? Is this right or wrong?

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Hamish
Hamish
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Some questions on the Dominated Convergence theorem

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now its is clear that $f_n\rightarrow 0$, but in the text I am using it says we can't apply the Dominated Convergence theorem as there is no function to dominate $f_n$, but I have trouble seeing that. To me the $f_n$'s are dominated by the constant function $1$. So I think can apply the Dominated Convergence theorem in some cases which depends on the choice measure where the constant functions are integrable. So if we are using the Lebesgue measure (on $\mathbb{R}$) then we can't use the Dominated Convergence theorem(and the text is right), but if are using a finite measure then will it be valid to use the Dominated Convergence theorem and we will then have $\lim\limits_n\int f_n\rightarrow0$. Also if our space was just a bounded interval and we have the Lebesgue measure (since it is now a finite measure on the internal), then it will it be fine to use the Dominated Convergence theorem on the $f_n$'s ? Is this right or wrong please let me know if am wrong and correct me. Thanks.