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Measure Theory and Functional Analysis




                    Notes          1.1 Differentiation and Integration


                                   1.1.1 Lipschitz Condition

                                   Definition: A function f defined on [a, b] is said to satisfy Lipschitz condition (or Lipschitzian
                                   function), if   a constant M > 0 s.t.

                                                    |f (x) – f (y)|  M |x – y|,   x, y   [a, b].

                                   1.1.2 Lebesgue Point of a Function

                                   Definition: A point x is said to be a Lebesgue point of the function f (t), if

                                                                 1  x n
                                                             Lim      |f(t) f(x)|dt  0.
                                                              h  0 h  x
                                   1.1.3 Covering in the Sense of Vitali


                                   Definition: A set E is said to be covered in the sense of Vitali by a family of intervals (may be open,
                                   closed or half open), M in which none is a singleton set, if every point of the set E is contained in
                                   some small interval of M i.e., for each x   E,   and   > 0, an interval I   M s.t. x  I and  (I)  .

                                   The family M is called the Vitali Cover of set E.

                                          Example: If E = {q : q is a rational number in the interval [a, b]}, then the family  I
                                                                                                             q i
                                                 1    1
                                   where  I   q    , q  , i   N is a vitali cover of [a, b].
                                          q i
                                                 i    i
                                   Vitali's Lemma

                                   Let E be a set of finite outer measure and M be a family of intervals which cover E in the sense of
                                   Vitali; then for a given   > 0, it is possible to find a finite family of disjoint intervals {I , k = 1, 2,
                                                                                                       k
                                   … n} of M, such that
                                                          n
                                                         
                                                   m * E    I  k  <  .
                                                          k 1
                                   Proof: Without any loss of generality, we assume that every interval of family M is a closed
                                   interval, because if not we replace each interval by its closure and observe that the set of end
                                   points of I , I , …… I  has measure zero.
                                           1  2    n
                                   [Due to this property some authors take family M of closed intervals in the definition of Vitali’s
                                   covering].
                                   Suppose 0 is an open set containing E s.t. m* (0) < m* (E) + 1 <   we assume that each interval in
                                   M is contained in 0, if this can be achieved by discarding the intervals of M extending beyond 0
                                   and still the family M will cover the set E in the sense of Vitali.
                                   Now we shall use the induction method to determine the sequence <I  : k = 1, 2, … n> of disjoint
                                                                                          k
                                   intervals of M as follows:






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