If $f_n$ $longrightarrow$ $f$ in measure and $g_n$ $longrightarrow$ $g$ in measure then show that $f_n +g_n$...











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(1)
If $f_n$ $longrightarrow$ $f$ in measure and $g_n$ $longrightarrow$ $g$ in measure then show that $f_n +g_n$ $longrightarrow$ $f+g$ in measure



(2)If $f_n longrightarrow theta$ in measure and $g_n longrightarrow theta$ in measure where $theta$ is the identically zero function on $Omega$ then show that $f_ng_n longrightarrow 0$



My attempt
For both cases I use the fact that $(Omega,Sigma,mu)$ is my measure space where everything is defined in the H.L Royden.



For (1) it follows from the relation $ {x in E |(f_n+g_n)(x)-(f+g)(x)}|geq epsilon$}



the above is a subset of the set $|f_n(x)-f(x)|geq epsilon/2$ obviously and thus is the



union of the set $|g_n(x)-g(x)|geq epsilon/2$ and hence the resukt holds.



Note here my measure is taken over an arbitrary set E. And this shows (1)



For (2) im drawing a blank and I don't have any idea how to even begin such a question. How do I 'handle' the zero function ?










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  • I'm a bit puzzled about the title of your question. Since you already solved this part of the problem it would be more natural to to put the 2nd part in the title of your question...
    – saz
    Nov 21 at 7:03















up vote
1
down vote

favorite












(1)
If $f_n$ $longrightarrow$ $f$ in measure and $g_n$ $longrightarrow$ $g$ in measure then show that $f_n +g_n$ $longrightarrow$ $f+g$ in measure



(2)If $f_n longrightarrow theta$ in measure and $g_n longrightarrow theta$ in measure where $theta$ is the identically zero function on $Omega$ then show that $f_ng_n longrightarrow 0$



My attempt
For both cases I use the fact that $(Omega,Sigma,mu)$ is my measure space where everything is defined in the H.L Royden.



For (1) it follows from the relation $ {x in E |(f_n+g_n)(x)-(f+g)(x)}|geq epsilon$}



the above is a subset of the set $|f_n(x)-f(x)|geq epsilon/2$ obviously and thus is the



union of the set $|g_n(x)-g(x)|geq epsilon/2$ and hence the resukt holds.



Note here my measure is taken over an arbitrary set E. And this shows (1)



For (2) im drawing a blank and I don't have any idea how to even begin such a question. How do I 'handle' the zero function ?










share|cite|improve this question
























  • I'm a bit puzzled about the title of your question. Since you already solved this part of the problem it would be more natural to to put the 2nd part in the title of your question...
    – saz
    Nov 21 at 7:03













up vote
1
down vote

favorite









up vote
1
down vote

favorite











(1)
If $f_n$ $longrightarrow$ $f$ in measure and $g_n$ $longrightarrow$ $g$ in measure then show that $f_n +g_n$ $longrightarrow$ $f+g$ in measure



(2)If $f_n longrightarrow theta$ in measure and $g_n longrightarrow theta$ in measure where $theta$ is the identically zero function on $Omega$ then show that $f_ng_n longrightarrow 0$



My attempt
For both cases I use the fact that $(Omega,Sigma,mu)$ is my measure space where everything is defined in the H.L Royden.



For (1) it follows from the relation $ {x in E |(f_n+g_n)(x)-(f+g)(x)}|geq epsilon$}



the above is a subset of the set $|f_n(x)-f(x)|geq epsilon/2$ obviously and thus is the



union of the set $|g_n(x)-g(x)|geq epsilon/2$ and hence the resukt holds.



Note here my measure is taken over an arbitrary set E. And this shows (1)



For (2) im drawing a blank and I don't have any idea how to even begin such a question. How do I 'handle' the zero function ?










share|cite|improve this question















(1)
If $f_n$ $longrightarrow$ $f$ in measure and $g_n$ $longrightarrow$ $g$ in measure then show that $f_n +g_n$ $longrightarrow$ $f+g$ in measure



(2)If $f_n longrightarrow theta$ in measure and $g_n longrightarrow theta$ in measure where $theta$ is the identically zero function on $Omega$ then show that $f_ng_n longrightarrow 0$



My attempt
For both cases I use the fact that $(Omega,Sigma,mu)$ is my measure space where everything is defined in the H.L Royden.



For (1) it follows from the relation $ {x in E |(f_n+g_n)(x)-(f+g)(x)}|geq epsilon$}



the above is a subset of the set $|f_n(x)-f(x)|geq epsilon/2$ obviously and thus is the



union of the set $|g_n(x)-g(x)|geq epsilon/2$ and hence the resukt holds.



Note here my measure is taken over an arbitrary set E. And this shows (1)



For (2) im drawing a blank and I don't have any idea how to even begin such a question. How do I 'handle' the zero function ?







measure-theory convergence






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edited Nov 21 at 10:13









saz

77.3k755119




77.3k755119










asked Nov 21 at 1:43









Jason Moore

607




607












  • I'm a bit puzzled about the title of your question. Since you already solved this part of the problem it would be more natural to to put the 2nd part in the title of your question...
    – saz
    Nov 21 at 7:03


















  • I'm a bit puzzled about the title of your question. Since you already solved this part of the problem it would be more natural to to put the 2nd part in the title of your question...
    – saz
    Nov 21 at 7:03
















I'm a bit puzzled about the title of your question. Since you already solved this part of the problem it would be more natural to to put the 2nd part in the title of your question...
– saz
Nov 21 at 7:03




I'm a bit puzzled about the title of your question. Since you already solved this part of the problem it would be more natural to to put the 2nd part in the title of your question...
– saz
Nov 21 at 7:03










1 Answer
1






active

oldest

votes

















up vote
2
down vote



accepted










Hints for (2):




  1. Let $(h_n)_{n in mathbb{N}}$ be a sequence of measurable functions. Prove that $h_n to 0$ in measure implies $h_n^2 to 0$ in measure.

  2. Use the first part of your problem and Step 1 to show that $(f_n+g_n)^2 to 0$ in measure and $(f_n-g_n)^2 to 0$ in measure.

  3. Use $$f_n cdot g_n = frac{1}{4} big( (f_n+g_n)^2- (f_n-g_n)^2 big)$$ and Step 1 to conclude that $f_n cdot g_n to 0$ in measure.






share|cite|improve this answer





















  • Thanks very much this makes more sense to me now!
    – Jason Moore
    Nov 21 at 14:27










  • @JasonMoore You are welcome.
    – saz
    Nov 21 at 14:41











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
2
down vote



accepted










Hints for (2):




  1. Let $(h_n)_{n in mathbb{N}}$ be a sequence of measurable functions. Prove that $h_n to 0$ in measure implies $h_n^2 to 0$ in measure.

  2. Use the first part of your problem and Step 1 to show that $(f_n+g_n)^2 to 0$ in measure and $(f_n-g_n)^2 to 0$ in measure.

  3. Use $$f_n cdot g_n = frac{1}{4} big( (f_n+g_n)^2- (f_n-g_n)^2 big)$$ and Step 1 to conclude that $f_n cdot g_n to 0$ in measure.






share|cite|improve this answer





















  • Thanks very much this makes more sense to me now!
    – Jason Moore
    Nov 21 at 14:27










  • @JasonMoore You are welcome.
    – saz
    Nov 21 at 14:41















up vote
2
down vote



accepted










Hints for (2):




  1. Let $(h_n)_{n in mathbb{N}}$ be a sequence of measurable functions. Prove that $h_n to 0$ in measure implies $h_n^2 to 0$ in measure.

  2. Use the first part of your problem and Step 1 to show that $(f_n+g_n)^2 to 0$ in measure and $(f_n-g_n)^2 to 0$ in measure.

  3. Use $$f_n cdot g_n = frac{1}{4} big( (f_n+g_n)^2- (f_n-g_n)^2 big)$$ and Step 1 to conclude that $f_n cdot g_n to 0$ in measure.






share|cite|improve this answer





















  • Thanks very much this makes more sense to me now!
    – Jason Moore
    Nov 21 at 14:27










  • @JasonMoore You are welcome.
    – saz
    Nov 21 at 14:41













up vote
2
down vote



accepted







up vote
2
down vote



accepted






Hints for (2):




  1. Let $(h_n)_{n in mathbb{N}}$ be a sequence of measurable functions. Prove that $h_n to 0$ in measure implies $h_n^2 to 0$ in measure.

  2. Use the first part of your problem and Step 1 to show that $(f_n+g_n)^2 to 0$ in measure and $(f_n-g_n)^2 to 0$ in measure.

  3. Use $$f_n cdot g_n = frac{1}{4} big( (f_n+g_n)^2- (f_n-g_n)^2 big)$$ and Step 1 to conclude that $f_n cdot g_n to 0$ in measure.






share|cite|improve this answer












Hints for (2):




  1. Let $(h_n)_{n in mathbb{N}}$ be a sequence of measurable functions. Prove that $h_n to 0$ in measure implies $h_n^2 to 0$ in measure.

  2. Use the first part of your problem and Step 1 to show that $(f_n+g_n)^2 to 0$ in measure and $(f_n-g_n)^2 to 0$ in measure.

  3. Use $$f_n cdot g_n = frac{1}{4} big( (f_n+g_n)^2- (f_n-g_n)^2 big)$$ and Step 1 to conclude that $f_n cdot g_n to 0$ in measure.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 21 at 7:01









saz

77.3k755119




77.3k755119












  • Thanks very much this makes more sense to me now!
    – Jason Moore
    Nov 21 at 14:27










  • @JasonMoore You are welcome.
    – saz
    Nov 21 at 14:41


















  • Thanks very much this makes more sense to me now!
    – Jason Moore
    Nov 21 at 14:27










  • @JasonMoore You are welcome.
    – saz
    Nov 21 at 14:41
















Thanks very much this makes more sense to me now!
– Jason Moore
Nov 21 at 14:27




Thanks very much this makes more sense to me now!
– Jason Moore
Nov 21 at 14:27












@JasonMoore You are welcome.
– saz
Nov 21 at 14:41




@JasonMoore You are welcome.
– saz
Nov 21 at 14:41


















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