Quick question about the term ‘integrable’
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1
down vote
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If a math textbook or paper just uses the term “integrable”, for example:
$$f(x) text{is integrable on} [a,b],$$
do we always assume this means Riemann integrable?
Thanks in advance!
real-analysis integration
asked Dec 1 at 20:51
MathIsLife12
549111
add a comment |
up vote
1
down vote
favorite
If a math textbook or paper just uses the term “integrable”, for example:
$$f(x) text{is integrable on} [a,b],$$
do we always assume this means Riemann integrable?
Thanks in advance!
real-analysis integration
asked Dec 1 at 20:51
MathIsLife12
549111
5
There's nothing certain about $f$ being integrable. It can be Riemann integrable, Lebesgue, Stjeltjes, and so on
– Jakobian
Dec 1 at 20:53
6
No, in general you cannot assume this. However, in most introductory books on real analysis it is usually taken to mean Riemann/Darboux integrability (they are equivalent).
– projectilemotion
Dec 1 at 20:57
write a function as $f(x)$ instead of just $f$ make me think that this is an elementary book, so probably it will mean that $f$ is Riemann integrable.
– Masacroso
Dec 1 at 21:26
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If a math textbook or paper just uses the term “integrable”, for example:
$$f(x) text{is integrable on} [a,b],$$
do we always assume this means Riemann integrable?
Thanks in advance!
real-analysis integration
asked Dec 1 at 20:51
MathIsLife12
549111
If a math textbook or paper just uses the term “integrable”, for example:
$$f(x) text{is integrable on} [a,b],$$
do we always assume this means Riemann integrable?
Thanks in advance!
real-analysis integration
real-analysis integration
asked Dec 1 at 20:51
MathIsLife12
549111
asked Dec 1 at 20:51
MathIsLife12
549111
asked Dec 1 at 20:51
MathIsLife12
549111
asked Dec 1 at 20:51
MathIsLife12
549111
asked Dec 1 at 20:51
MathIsLife12
549111
549111
5
There's nothing certain about $f$ being integrable. It can be Riemann integrable, Lebesgue, Stjeltjes, and so on
– Jakobian
Dec 1 at 20:53
6
No, in general you cannot assume this. However, in most introductory books on real analysis it is usually taken to mean Riemann/Darboux integrability (they are equivalent).
– projectilemotion
Dec 1 at 20:57
write a function as $f(x)$ instead of just $f$ make me think that this is an elementary book, so probably it will mean that $f$ is Riemann integrable.
– Masacroso
Dec 1 at 21:26
add a comment |
5
There's nothing certain about $f$ being integrable. It can be Riemann integrable, Lebesgue, Stjeltjes, and so on
– Jakobian
Dec 1 at 20:53
6
No, in general you cannot assume this. However, in most introductory books on real analysis it is usually taken to mean Riemann/Darboux integrability (they are equivalent).
– projectilemotion
Dec 1 at 20:57
write a function as $f(x)$ instead of just $f$ make me think that this is an elementary book, so probably it will mean that $f$ is Riemann integrable.
– Masacroso
Dec 1 at 21:26
5
5
There's nothing certain about $f$ being integrable. It can be Riemann integrable, Lebesgue, Stjeltjes, and so on
– Jakobian
Dec 1 at 20:53
There's nothing certain about $f$ being integrable. It can be Riemann integrable, Lebesgue, Stjeltjes, and so on
– Jakobian
Dec 1 at 20:53
6
6
No, in general you cannot assume this. However, in most introductory books on real analysis it is usually taken to mean Riemann/Darboux integrability (they are equivalent).
– projectilemotion
Dec 1 at 20:57
No, in general you cannot assume this. However, in most introductory books on real analysis it is usually taken to mean Riemann/Darboux integrability (they are equivalent).
– projectilemotion
Dec 1 at 20:57
write a function as $f(x)$ instead of just $f$ make me think that this is an elementary book, so probably it will mean that $f$ is Riemann integrable.
– Masacroso
Dec 1 at 21:26
write a function as $f(x)$ instead of just $f$ make me think that this is an elementary book, so probably it will mean that $f$ is Riemann integrable.
– Masacroso
Dec 1 at 21:26
add a comment |
1 Answer
1
active
oldest
votes
up vote
4
down vote
Depends on the context. For instance, the function
$f: [a,b] longrightarrow mathbb R$ defined by
$$f(x)=begin{cases}1 &text{ if }x in mathbb Q \0&text{ otherwise}end{cases}$$
is Lebesgue integrable but not Riemann integrable.
answered Dec 1 at 20:59
Matt A Pelto
2,338620
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Depends on the context. For instance, the function
$f: [a,b] longrightarrow mathbb R$ defined by
$$f(x)=begin{cases}1 &text{ if }x in mathbb Q \0&text{ otherwise}end{cases}$$
is Lebesgue integrable but not Riemann integrable.
answered Dec 1 at 20:59
Matt A Pelto
2,338620
add a comment |
up vote
4
down vote
Depends on the context. For instance, the function
$f: [a,b] longrightarrow mathbb R$ defined by
$$f(x)=begin{cases}1 &text{ if }x in mathbb Q \0&text{ otherwise}end{cases}$$
is Lebesgue integrable but not Riemann integrable.
answered Dec 1 at 20:59
Matt A Pelto
2,338620
add a comment |
up vote
4
down vote
up vote
4
down vote
Depends on the context. For instance, the function
$f: [a,b] longrightarrow mathbb R$ defined by
$$f(x)=begin{cases}1 &text{ if }x in mathbb Q \0&text{ otherwise}end{cases}$$
is Lebesgue integrable but not Riemann integrable.
answered Dec 1 at 20:59
Matt A Pelto
2,338620
Depends on the context. For instance, the function
$f: [a,b] longrightarrow mathbb R$ defined by
$$f(x)=begin{cases}1 &text{ if }x in mathbb Q \0&text{ otherwise}end{cases}$$
is Lebesgue integrable but not Riemann integrable.
answered Dec 1 at 20:59
Matt A Pelto
2,338620
answered Dec 1 at 20:59
Matt A Pelto
2,338620
answered Dec 1 at 20:59
Matt A Pelto
2,338620
answered Dec 1 at 20:59
Matt A Pelto
2,338620
2,338620
add a comment |
add a comment |
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5
There's nothing certain about $f$ being integrable. It can be Riemann integrable, Lebesgue, Stjeltjes, and so on
– Jakobian
Dec 1 at 20:53
6
No, in general you cannot assume this. However, in most introductory books on real analysis it is usually taken to mean Riemann/Darboux integrability (they are equivalent).
– projectilemotion
Dec 1 at 20:57
write a function as $f(x)$ instead of just $f$ make me think that this is an elementary book, so probably it will mean that $f$ is Riemann integrable.
– Masacroso
Dec 1 at 21:26