Lebesgue Fundamental Theorem of Calculus problems
I have a couple problems related to the Lebesgue FTC that I'm having a hard time solving:
- If $F:mathbb Rto mathbb Cin$NBV, show that there is a null Borel set $Nsubsetmathbb R$ s.t. if $[a,b]cap N=0,$ then $$F(b)-F(a)=int_a^bF'(t)dt$$.
- Show that if $F:mathbb R to mathbb R$ is increasing and $-infty<a<b<infty$, then $$F(b)-F(a)geq int_a^bF'(t)dt$$
For both of the problems, I'm kind of at an impasse and just running around in circles. I would greatly appreciate some help on solving these problems.
real-analysis integration measure-theory lebesgue-integral
asked Nov 30 at 1:47
Mog
549
add a comment |
I have a couple problems related to the Lebesgue FTC that I'm having a hard time solving:
- If $F:mathbb Rto mathbb Cin$NBV, show that there is a null Borel set $Nsubsetmathbb R$ s.t. if $[a,b]cap N=0,$ then $$F(b)-F(a)=int_a^bF'(t)dt$$.
- Show that if $F:mathbb R to mathbb R$ is increasing and $-infty<a<b<infty$, then $$F(b)-F(a)geq int_a^bF'(t)dt$$
For both of the problems, I'm kind of at an impasse and just running around in circles. I would greatly appreciate some help on solving these problems.
real-analysis integration measure-theory lebesgue-integral
asked Nov 30 at 1:47
Mog
549
What is NBV? Show us your work.
– Sean Roberson
Nov 30 at 1:51
@SeanRoberson NBV is the set of functions of bounded variation that are right continuous, and equals $0$ at $-infty$. I'm not really sure where to begin on these two.
– Mog
Nov 30 at 1:55
add a comment |
I have a couple problems related to the Lebesgue FTC that I'm having a hard time solving:
- If $F:mathbb Rto mathbb Cin$NBV, show that there is a null Borel set $Nsubsetmathbb R$ s.t. if $[a,b]cap N=0,$ then $$F(b)-F(a)=int_a^bF'(t)dt$$.
- Show that if $F:mathbb R to mathbb R$ is increasing and $-infty<a<b<infty$, then $$F(b)-F(a)geq int_a^bF'(t)dt$$
For both of the problems, I'm kind of at an impasse and just running around in circles. I would greatly appreciate some help on solving these problems.
real-analysis integration measure-theory lebesgue-integral
asked Nov 30 at 1:47
Mog
549
I have a couple problems related to the Lebesgue FTC that I'm having a hard time solving:
- If $F:mathbb Rto mathbb Cin$NBV, show that there is a null Borel set $Nsubsetmathbb R$ s.t. if $[a,b]cap N=0,$ then $$F(b)-F(a)=int_a^bF'(t)dt$$.
- Show that if $F:mathbb R to mathbb R$ is increasing and $-infty<a<b<infty$, then $$F(b)-F(a)geq int_a^bF'(t)dt$$
For both of the problems, I'm kind of at an impasse and just running around in circles. I would greatly appreciate some help on solving these problems.
real-analysis integration measure-theory lebesgue-integral
real-analysis integration measure-theory lebesgue-integral
asked Nov 30 at 1:47
Mog
549
asked Nov 30 at 1:47
Mog
549
asked Nov 30 at 1:47
Mog
549
asked Nov 30 at 1:47
Mog
549
asked Nov 30 at 1:47
Mog
549
549
What is NBV? Show us your work.
– Sean Roberson
Nov 30 at 1:51
@SeanRoberson NBV is the set of functions of bounded variation that are right continuous, and equals $0$ at $-infty$. I'm not really sure where to begin on these two.
– Mog
Nov 30 at 1:55
add a comment |
What is NBV? Show us your work.
– Sean Roberson
Nov 30 at 1:51
@SeanRoberson NBV is the set of functions of bounded variation that are right continuous, and equals $0$ at $-infty$. I'm not really sure where to begin on these two.
– Mog
Nov 30 at 1:55
What is NBV? Show us your work.
– Sean Roberson
Nov 30 at 1:51
What is NBV? Show us your work.
– Sean Roberson
Nov 30 at 1:51
@SeanRoberson NBV is the set of functions of bounded variation that are right continuous, and equals $0$ at $-infty$. I'm not really sure where to begin on these two.
– Mog
Nov 30 at 1:55
@SeanRoberson NBV is the set of functions of bounded variation that are right continuous, and equals $0$ at $-infty$. I'm not really sure where to begin on these two.
– Mog
Nov 30 at 1:55
add a comment |
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What is NBV? Show us your work.
– Sean Roberson
Nov 30 at 1:51
@SeanRoberson NBV is the set of functions of bounded variation that are right continuous, and equals $0$ at $-infty$. I'm not really sure where to begin on these two.
– Mog
Nov 30 at 1:55