Page 77 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
2
Example: If <f > is a sequence of functions belonging to L (a, b) and also f L (a, b) and
2
n
Lim f – f = 0, then prove that
n 2
b b
2
2
f dx Lim f dx
n
a a
Solution: By Minkowski’s inequality, we get
f f f – f
n 2 2 n 2
Lim f f Lim f – f = 0
n 2 n 2

Lim f f = 0 Lim f = f
n 2 n 2 2
1/2 1/2
b b b b
2
2
2
2
Lim (f ) dx f dx Lim f dx f dx .
n n
a a a a
6.2 Summary
p
 The class of all measurable function f (x) is known as L space over [a, b], if Lebesgue
integrable over [a, b] for each p exists, 0 < p < .
p
p
 If f and g L (1 p ), then f + g L and f + g f + g .
p p p
6.3 Keywords
L -space: The class of all measurable functions f (x) is known as L -space over [a, b], if Lebesgue-
p
p
integrable over [a, b] for each exists, 0 < p < , i.e.,
b
p
|f| dx , (p 0)
a
p
and is denoted by L [a, b].
Minkowski Inequality in Integral Form: Suppose f :  ×   is Lebesgue measurable and 1
p < . Then
1/p 1/p
p
h(x, y) dy dx h(x, y) dx dy

Minkowski Inequality: Minkowski inequality establishes that the L spaces are normed vector
p
p
spaces. Let S be a measure space, let 1 p and let f and g be elements of L (s). Then f + g is in
p
L (s), we have the triangle inequality
f + g f + g
p p p
with equality for 1 < p < if and only if f and g are positively linearly dependent.

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