On the history of Haar measureWhat group theoretic results were known for several special cases before the general definition of a group was established?Request for good resources on 'history of infinity' topicsHistory of measure theoryOrigins and history of branched coveringWho first wrote down $S^6$'s standard almost complex structure? And who first proved that it is not integrable?History of BraidsHistory of group theory character tables (as used in physics and chemistry)How did the integer degrees angles counting being first adopted in geometry and mathematics?What are some good books that interweave the history of math and art from renaissance onward?Material on the History of Mathematical Spaces
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On the history of Haar measure
What group theoretic results were known for several special cases before the general definition of a group was established?Request for good resources on 'history of infinity' topicsHistory of measure theoryOrigins and history of branched coveringWho first wrote down $S^6$'s standard almost complex structure? And who first proved that it is not integrable?History of BraidsHistory of group theory character tables (as used in physics and chemistry)How did the integer degrees angles counting being first adopted in geometry and mathematics?What are some good books that interweave the history of math and art from renaissance onward?Material on the History of Mathematical Spaces
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Haar measure is a well-known concept in measure theory.
Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn.
I am looking for a good reference on the history of Haar measure.
mathematics topology group-theory
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
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add a comment |
$begingroup$
Haar measure is a well-known concept in measure theory.
Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn.
I am looking for a good reference on the history of Haar measure.
mathematics topology group-theory
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
$endgroup$
add a comment |
$begingroup$
Haar measure is a well-known concept in measure theory.
Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn.
I am looking for a good reference on the history of Haar measure.
mathematics topology group-theory
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
$endgroup$
Haar measure is a well-known concept in measure theory.
Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn.
I am looking for a good reference on the history of Haar measure.
mathematics topology group-theory
mathematics topology group-theory
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
asked Jul 23 at 18:22
Neil hawkingNeil hawking
1361 bronze badge
1361 bronze badge
add a comment |
add a comment |
2 Answers
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Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt:
"Invariant integration on one or another special class of groups has
long been known and used. A detailed computation of the invariant
integral on $mathfrakSD(n)$ was given in 1897 by HURWITZ [1]. SCHUR and
FROBEKius in the years 1900-1920 made frequent use of averages over finite groups; for references, see the notes in WEYL [3]. SCHUR in [ 1]
computed and applied intensively the invariant integrals for $mathfrakSD(n)$
and $mathfrakD(n)$. WEYL in [1] computed the invariant integrals for $mathfrakU(n)$,
$mathfrakSD(n)$, the unitary subgroup of the symplectic group, and [more or less
explicitly] for certain other compact Lie groups. WEYL and PETER in [1]
showed the existence of an invariant integral for any compact Lie group.
The decisive step in founding modern harmonic analysis was taken by
A. HAAR [3] in 1933. He proved directly the existence [but not the
uniqueness] of left Haar measure on a locally compact group with a
countable open basis. His construction was reformulated in t erms of
linear functionals and extended to arbitrary locally compact groups by
A. WEIL [1], [2], and [4], pp. 33 -38. KAKUTANI [2] pointed out also
that HAAR's construction can be extended to all locally compact groups.
Theorem ( 15. S) as stated is thus due to WEIL. The proof we present
is due to H. CARTAN [1].
For an arbitrary compact group G, VON NEUMANN [5] proved the
existence and uniqueness of the Haar integral, as well as its two-sided
and inversion invariance. In [6], VON NEUMANN proved the uniqueness
of left Haar measure for locally compact G with a countable open basis;
a special case was also established by Sz.-NAGY [1 ]. WEIL [ 4], pp. 37-38,
proved the uniqueness of the left Haar integral for all locally compact
groups."
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
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Try these references:
Section 7.5 of History of Topology, edited by I. M. James.
Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology, volume 3, edited by C.E. All and R. Lowen
answered Jul 23 at 18:31
lhflhf
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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active
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active
oldest
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$begingroup$
Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt:
"Invariant integration on one or another special class of groups has
long been known and used. A detailed computation of the invariant
integral on $mathfrakSD(n)$ was given in 1897 by HURWITZ [1]. SCHUR and
FROBEKius in the years 1900-1920 made frequent use of averages over finite groups; for references, see the notes in WEYL [3]. SCHUR in [ 1]
computed and applied intensively the invariant integrals for $mathfrakSD(n)$
and $mathfrakD(n)$. WEYL in [1] computed the invariant integrals for $mathfrakU(n)$,
$mathfrakSD(n)$, the unitary subgroup of the symplectic group, and [more or less
explicitly] for certain other compact Lie groups. WEYL and PETER in [1]
showed the existence of an invariant integral for any compact Lie group.
The decisive step in founding modern harmonic analysis was taken by
A. HAAR [3] in 1933. He proved directly the existence [but not the
uniqueness] of left Haar measure on a locally compact group with a
countable open basis. His construction was reformulated in t erms of
linear functionals and extended to arbitrary locally compact groups by
A. WEIL [1], [2], and [4], pp. 33 -38. KAKUTANI [2] pointed out also
that HAAR's construction can be extended to all locally compact groups.
Theorem ( 15. S) as stated is thus due to WEIL. The proof we present
is due to H. CARTAN [1].
For an arbitrary compact group G, VON NEUMANN [5] proved the
existence and uniqueness of the Haar integral, as well as its two-sided
and inversion invariance. In [6], VON NEUMANN proved the uniqueness
of left Haar measure for locally compact G with a countable open basis;
a special case was also established by Sz.-NAGY [1 ]. WEIL [ 4], pp. 37-38,
proved the uniqueness of the left Haar integral for all locally compact
groups."
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
$endgroup$
add a comment |
$begingroup$
Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt:
"Invariant integration on one or another special class of groups has
long been known and used. A detailed computation of the invariant
integral on $mathfrakSD(n)$ was given in 1897 by HURWITZ [1]. SCHUR and
FROBEKius in the years 1900-1920 made frequent use of averages over finite groups; for references, see the notes in WEYL [3]. SCHUR in [ 1]
computed and applied intensively the invariant integrals for $mathfrakSD(n)$
and $mathfrakD(n)$. WEYL in [1] computed the invariant integrals for $mathfrakU(n)$,
$mathfrakSD(n)$, the unitary subgroup of the symplectic group, and [more or less
explicitly] for certain other compact Lie groups. WEYL and PETER in [1]
showed the existence of an invariant integral for any compact Lie group.
The decisive step in founding modern harmonic analysis was taken by
A. HAAR [3] in 1933. He proved directly the existence [but not the
uniqueness] of left Haar measure on a locally compact group with a
countable open basis. His construction was reformulated in t erms of
linear functionals and extended to arbitrary locally compact groups by
A. WEIL [1], [2], and [4], pp. 33 -38. KAKUTANI [2] pointed out also
that HAAR's construction can be extended to all locally compact groups.
Theorem ( 15. S) as stated is thus due to WEIL. The proof we present
is due to H. CARTAN [1].
For an arbitrary compact group G, VON NEUMANN [5] proved the
existence and uniqueness of the Haar integral, as well as its two-sided
and inversion invariance. In [6], VON NEUMANN proved the uniqueness
of left Haar measure for locally compact G with a countable open basis;
a special case was also established by Sz.-NAGY [1 ]. WEIL [ 4], pp. 37-38,
proved the uniqueness of the left Haar integral for all locally compact
groups."
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
$endgroup$
add a comment |
$begingroup$
Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt:
"Invariant integration on one or another special class of groups has
long been known and used. A detailed computation of the invariant
integral on $mathfrakSD(n)$ was given in 1897 by HURWITZ [1]. SCHUR and
FROBEKius in the years 1900-1920 made frequent use of averages over finite groups; for references, see the notes in WEYL [3]. SCHUR in [ 1]
computed and applied intensively the invariant integrals for $mathfrakSD(n)$
and $mathfrakD(n)$. WEYL in [1] computed the invariant integrals for $mathfrakU(n)$,
$mathfrakSD(n)$, the unitary subgroup of the symplectic group, and [more or less
explicitly] for certain other compact Lie groups. WEYL and PETER in [1]
showed the existence of an invariant integral for any compact Lie group.
The decisive step in founding modern harmonic analysis was taken by
A. HAAR [3] in 1933. He proved directly the existence [but not the
uniqueness] of left Haar measure on a locally compact group with a
countable open basis. His construction was reformulated in t erms of
linear functionals and extended to arbitrary locally compact groups by
A. WEIL [1], [2], and [4], pp. 33 -38. KAKUTANI [2] pointed out also
that HAAR's construction can be extended to all locally compact groups.
Theorem ( 15. S) as stated is thus due to WEIL. The proof we present
is due to H. CARTAN [1].
For an arbitrary compact group G, VON NEUMANN [5] proved the
existence and uniqueness of the Haar integral, as well as its two-sided
and inversion invariance. In [6], VON NEUMANN proved the uniqueness
of left Haar measure for locally compact G with a countable open basis;
a special case was also established by Sz.-NAGY [1 ]. WEIL [ 4], pp. 37-38,
proved the uniqueness of the left Haar integral for all locally compact
groups."
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
$endgroup$
Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt:
"Invariant integration on one or another special class of groups has
long been known and used. A detailed computation of the invariant
integral on $mathfrakSD(n)$ was given in 1897 by HURWITZ [1]. SCHUR and
FROBEKius in the years 1900-1920 made frequent use of averages over finite groups; for references, see the notes in WEYL [3]. SCHUR in [ 1]
computed and applied intensively the invariant integrals for $mathfrakSD(n)$
and $mathfrakD(n)$. WEYL in [1] computed the invariant integrals for $mathfrakU(n)$,
$mathfrakSD(n)$, the unitary subgroup of the symplectic group, and [more or less
explicitly] for certain other compact Lie groups. WEYL and PETER in [1]
showed the existence of an invariant integral for any compact Lie group.
The decisive step in founding modern harmonic analysis was taken by
A. HAAR [3] in 1933. He proved directly the existence [but not the
uniqueness] of left Haar measure on a locally compact group with a
countable open basis. His construction was reformulated in t erms of
linear functionals and extended to arbitrary locally compact groups by
A. WEIL [1], [2], and [4], pp. 33 -38. KAKUTANI [2] pointed out also
that HAAR's construction can be extended to all locally compact groups.
Theorem ( 15. S) as stated is thus due to WEIL. The proof we present
is due to H. CARTAN [1].
For an arbitrary compact group G, VON NEUMANN [5] proved the
existence and uniqueness of the Haar integral, as well as its two-sided
and inversion invariance. In [6], VON NEUMANN proved the uniqueness
of left Haar measure for locally compact G with a countable open basis;
a special case was also established by Sz.-NAGY [1 ]. WEIL [ 4], pp. 37-38,
proved the uniqueness of the left Haar integral for all locally compact
groups."
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
answered Jul 23 at 20:41
ConifoldConifold
38.3k1 gold badge61 silver badges134 bronze badges
38.3k1 gold badge61 silver badges134 bronze badges
add a comment |
add a comment |
$begingroup$
Try these references:
Section 7.5 of History of Topology, edited by I. M. James.
Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology, volume 3, edited by C.E. All and R. Lowen
answered Jul 23 at 18:31
lhflhf
3011 silver badge4 bronze badges
$endgroup$
add a comment |
$begingroup$
Try these references:
Section 7.5 of History of Topology, edited by I. M. James.
Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology, volume 3, edited by C.E. All and R. Lowen
answered Jul 23 at 18:31
lhflhf
3011 silver badge4 bronze badges
$endgroup$
add a comment |
$begingroup$
Try these references:
Section 7.5 of History of Topology, edited by I. M. James.
Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology, volume 3, edited by C.E. All and R. Lowen
answered Jul 23 at 18:31
lhflhf
3011 silver badge4 bronze badges
$endgroup$
Try these references:
Section 7.5 of History of Topology, edited by I. M. James.
Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology, volume 3, edited by C.E. All and R. Lowen
answered Jul 23 at 18:31
lhflhf
3011 silver badge4 bronze badges
answered Jul 23 at 18:31
lhflhf
3011 silver badge4 bronze badges
answered Jul 23 at 18:31
lhflhf
3011 silver badge4 bronze badges
answered Jul 23 at 18:31
lhflhf
3011 silver badge4 bronze badges
3011 silver badge4 bronze badges
add a comment |
add a comment |
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