Page 286 - DMTH401_REAL ANALYSIS

Real Analysis

Notes where the convergence is convergence in measure with respect to . In 1988, Tišer showed that if
s 2
s 2 i 1 £ i
+
i a
for some a > 5/2, then

1 f(y)d (y) ¾¾¾ f(x)

 (B (x)) ò r B (x) r 0
r
p
for -almost all x and all f  L (H, ; R), p > 1.
As of 2007, it is still an open question whether there exists an infinite-dimensional Gaussian
measure  on a separable Hilbert space H so that, for all f  L (H, ; R),
1
1

lim ò f(y)d (y) = f(x)
r  (B (x)) r B (x)
0
r
for -almost all x  H. However, it is conjectured that no such measure exists, since the s would
i
have to decay very rapidly.
æ 1 ö
a
Example: If a ¹ 0, ( ) arctan ç ÷
=
f
è ø
a
a
The function is not continuous at the point (x, a) = (0, 0) and the function f(a) has a
2
x + a 2
p p
+
-
discontinuity a = 0, because f(a) approaches + as a  0 and approaches - as a  0 .
2 2
1 a
If we now differentiate f(a) = ò 0 x + a 2 dx with respect to a under the integral sign, we get
2
1
2
d 1 x - a 2 x 1
a
f ( ) = ò dx = - = - which is, of course, true for all values of a except
2
2
da 0 x + a 2 x + a 2 1 + a 2
0
a= 0.
Example: The principle of differentiating under the integral sign may sometimes be
used to evaluate a definite integral.
p
-
a
Consider integrating ( )f a = 0 ò ln(1 2 cos(x) + a 2 )dx (for a > 1)
Now,
+
d p - 2 cos(x) 2a
f ( ) = ò dx
a
da 0 2 cos(x)- a + a 2
1 pæ (1 - a ) 2 ö
= 0 ç ò 1 - 2 ÷ dx
a è 1 2 cos(x) + a ø
a
-
p
p 2 ì æ 1 + a æ x ö ö ü
= - í arctan ç × tan ç ÷÷ ý
a a î è 1 - a è 2 øø 0 þ

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