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Is 'LUSIN THEOREM' true in infinite measure case?

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Benjamin

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Jul 16, 2005, 11:59:49 AM7/16/05
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In my book Lusin theorem is stated as following:

Let f be a measurable function defined on E. Then for each positive
epsilon, there exists a closed set F in E such that m(E-F)<epsilon and
f is continuous on F.

There is no restriction on E. I've started reading this book from the
first chapter so I'm sure set E has no restriction. Is it stil true if
m(E) is infinite? I found several different version of Lusion theorem.
One of it was in folland's REAL ANALYSIS book and E is just compact
set. Others have m(E) finite case. Thank you for any help.

José Carlos Santos

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Jul 16, 2005, 1:15:16 PM7/16/05
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Benjamin wrote:

> Let f be a measurable function defined on E. Then for each positive
> epsilon, there exists a closed set F in E such that m(E-F)<epsilon and
> f is continuous on F.

My guess is that you forgot to add that the image of f does not contain
+oo or -oo.

> There is no restriction on E. I've started reading this book from the
> first chapter so I'm sure set E has no restriction. Is it stil true if
> m(E) is infinite? I found several different version of Lusion theorem.
> One of it was in folland's REAL ANALYSIS book and E is just compact
> set. Others have m(E) finite case. Thank you for any help.

Suppose that E = R. Then, if f:R --> R is measurable and if epsilon > r,
choose some family (a_n)_{n in Z} of non-negative real numbers whose
sum is equal to 1. For each n in Z, let F_n be a closed subset of
[n,n + 1] such that the restriction of f to F_n is continuous and that
m([n,n + 1]\F) < epsilon x a_n. The the union F of all F_n is a closed
subset of R, f is continuous at all points of F (except, perhaps, when
those points are integers, but Z has measure 0) and m(R\F) < epsilon.

You can do the same thing in R^n. I don't know about the general cases.
Which topological hypothesis do you have concerning E. My guess is that
it is locally compact.

I hope that this helps.

Best regards,

Jose Carlos Santos

Dave L. Renfro

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Jul 17, 2005, 1:09:45 AM7/17/05
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Benjamin wrote:

I'm going to use your question as a launching pad
for a short essay that others might be interested in.
It concerns some things I was looking into this past
December that relate to Lusin's theorem.

LUSIN'S THEOREM (no frills version): Let f:R --> R
be measurable and epsilon > 0. Then there exists a
measurable set E such that measure(R-E) < epsilon
and the restriction of f to E is a continuous
function from E into R.

Note that we're talking about the _restriction_ of f
to E being continuous, not f itself being continuous
at each point of E. The characteristic function of
the rationals isn't continuous at a single point,
but after the removal of just countably many points
(thus, "measure(R-E) < epsilon" is satisfied in a
very strong way), we get a constant function (thus,
a function that is continuous in a very strong way).

FRILL 1: In the above, we can choose E to be closed.
In fact, we can choose E to be a perfect nowhere
dense set, and I believe this was the form it was
originally proved.

FRILL 2: In Frill 1 we can find a continuous g:R --> R
such that g(x) = f(x) for all x in E. This is because
we can extend any continuous function defined on a
closed subset of R to a continuous function defined
on all of R.

REMARK 1: Lusin's theorem fails for epsilon = 0.
(Consider the characteristic function of a perfect
nowhere dense set with positive measure.)

REMARK 2: Any function f:R --> R (not assumed measurable)
such that Lusin's theorem holds for all measurable sets
E (or even just all perfect nowhere dense sets E) must
be measurable. That is, the converse of Lusin's theorem
holds, and hence the "Lusin property" characterizes the
measurability of functions.

NEAT APPLICATION: If f:R --> R is unbounded on every
set of positive measure (or even on every perfect set
of positive measure), then f is not measurable. Note
that being unbounded on every _interval_ implies being
discontinuous at every point. (Hence, no such function
can be Baire one. However, there are Baire two functions
that are unbounded on every interval.)

Incidentally, Henry Blumberg proved in 1922 that given
an arbitrary f:R --> R, there exists a countable dense
subset D of R such that the restriction of f to D is
continuous (Blumberg, "New properties of all real
functions", Transactions of the American Mathematical
Society 24 (1922), 113-128). In particular, there
exists an infinite subset D such that the restriction
of f to D is continuous. On the other hand, Sierpinski
and Zygmund proved in 1923 that there exists a function
f:R --> R such that every restriction of f to a set
of cardinality c is discontinuous ("Sur une fonction
qui est discontinue sur tout ensemble de puissance du
continu", Fundamenta Mathematicae 4 (1923), 316-318).

APPLICATION OF THE APPLICATION: One can show that any
nonlinear function satisfying f(x+y) = f(x) + f(y)
for all x,y in R is unbounded in every interval.
Using the fact that if E has positive measure,
then {x-y: x,y in E} contains an interval, it is
not difficult to now show that any nonlinear additive
function is unbounded on every set of positive measure,
and hence is nonmeasurable. In fact, any such function
will also majorize every measurable function on every
set of positive measure. (Being unbounded just means
it majorizes any constant function.)

I pointed out above that Lusin's theorem fails if
epsilon = 0. However, if we weaken "continuous" to
"Baire one" (a pointwise limit of continuous functions),
then we can get an epsilon = 0 version. Although we
can't get E to be closed (see below), we can still
get E to be F_sigma (a countable union of closed sets).

BAIRE ONE VERSION OF LUSIN'S THEOREM: Let f:R --> R
be measurable. Then there exists an F_sigma set E such
that measure(R-E) = 0 and the restriction of f to E
is a Baire one function on E.

REMARK 3: The analog of Frill 2 above fails. There
exist measurable functions f:R --> R that are not
almost everywhere equal to any Baire one function
g:R --> R. (Consider the characteristic function of
a set such that both the set and its complement has
a positive measure intersection with every interval.
Oxtoby's book "Measure and Category", 2'nd edition,
p. 37 gives a very nice construction of such a set
that also happens to be F_sigma. Rudin gives the
same construction in "Well-distributed measurable
sets", American Mathematical Monthly 90 (1983),
41-42.)

Apparently, the place where things break down when
we try to prove a Baire one "epsilon = 0" version
of Frill 2 is that if E is F_sigma, then not every
Baire one function f:E --> R can be extended to all
of R. (Baire one functions on G_delta sets can be
extended to Baire one functions on all of R,
by the way.) There doesn't seem to be much in
the literature concerning extending Baire one
functions, and I'd welcome any references that
someone might know of. About the only relevant
reference I'm aware of is a recent manuscript
by Kalenda and Spurny titled "Extending Baire-one
functions on topological spaces". However, their
focus is on how various topological assumptions
affect things rather than on a detailed analysis
of the situation for real-valued functions of
a real variable.

REMARK 4: The analog of Frill 2 does hold if we weaken
"Baire one" to "Baire two". That is, if f:R --> R
is measurable, then there exists an F_sigma set E
and a Baire two function g:R --> R such that
measure (R-E) = 0 and f(x) = g(x) for all x in E.
In fact, there exists functions g_1 and g_2 that
are C_UL and C_LU in Young's classification [1],
respectively, such that g_1 \leq f \leq g_2 and
g_1 = g_2 almost everywhere. This result is often
called the Vitali-Caratheodory theorem. I don't have
many references at my finger tips right now, but a
fairly good treatment can be found on pp. 144-147
of Hahn/Rosenthal's 1948 book "Set Functions", and
Young's own version appears on pp. 31-32 of his
paper "On a new method in the theory of integration",
Proceedings of the London Mathematical Society (2) 9
(1911), 15-50.

[1] g belongs to C_L means there exists a sequence
{f_n} of continuous functions such that
f_1 \leq f_2 \leq f_3 ... (\leq means "less
than or equal to") and {f_n} converges
pointwise to g. In short, g is an increasing
pointwise limit of continuous functions. C_U
consists of decreasing pointwise limits of
continuous functions. If g is bounded, then
g is C_L iff g is lower semicontinuous and
g is C_U iff g is upper semicontinuous. The
"only if" halves are true even if g is not
bounded, and so if g is both C_L and C_U,
g will be continuous. C_LU consists of
decreasing pointwise limits of C_L functions,
and similarly for C_UL. Young proved (pp. 23-24
of his 1911 paper I cite) that the collection
of Baire one functions is the intersection
of the C_LU and C_UL collections. I don't
remember if boundedness is needed for this
last result, and because it's so late right
now, I'm not going to look it up. But I do
know that aside from boundedness issues,
the Young hierarchy continues to refine the
Borel hierarchy ... Baire two functions are
the intersection of the C_LUL and C_ULU
collections, and so on (even transfinitely).
There's not much in the literature about the
Young hierarchy (Hahn's text from the 1920's
is probably the single best source), but one
paper that does discuss it some is Michal
Morayne, "Algebras of Borel measurable
functions", Fundamenta Mathematicae 141 (1992),
229-242. In fact, Morayne studies a refinement
that involves three or four sublevels inserted
between each of the Young levels.

Dave L. Renfro

José Carlos Santos

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Jul 17, 2005, 3:01:57 AM7/17/05
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Dave L. Renfro wrote:

> I'm going to use your question as a launching pad
> for a short essay that others might be interested in.
> It concerns some things I was looking into this past
> December that relate to Lusin's theorem.

If there was a sci.math Hall of Fame, your essay would belong
there.

Ross A. Finlayson

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May 6, 2020, 1:48:38 PM5/6/20
to
This is an interesting point, from 2005,

https://groups.google.com/d/msg/sci.math/TBLfQglZEIA/kQu17saRBmgJ

and about Blumberg 1922,
"(Blumberg, "New properties of all real functions",
Transactions of the American Mathematical Society 24 (1922), 113-128)"

"The new properties are of two kinds; descriptive and metric...".

1922: not so new any-more.

It's interesting that in physics where quantum mechanics and
relativity need to get together, that in mathematics it's
the descriptive and metric, in similar analogy, two in a
sense inchoate branches, for unification (foundations).

That quantum mechanics and relativity are both successful
theories that don't agree in extremes, is how they do,
what they really "are", how mathematics "agrees" and
physics "agrees".

Continuum mechanics....

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