Inherent Pitfall of Lebesgue Integration?I am studying Real Analysis with Royden's Book. I noticed that for a function f differentiable almost everywhere on [a,b] and f′ integrable over [a,b], it does not imply that f ( x ) = ∫ [ a , x ] f ′ ( t ) d t + f ( a ) .. Take Cantor function f as a counter example, since 1 = f ( 1 ) &gt; ∫ [ 0 , 1 ] f ′ ( t ) d t + f ( 0 ) = 0 ,, because ∀ x ∈ [ 0 , 1 ]. However, does it mean that Lebesgue integral has some inherent pitfall in the definition of integration? In a sense that it fails to restore the original picture of a function in a set of measure zero after differentiation. Is that any better (or more general) design of integration? so that we can guarantee f ( x ) = ∫ [ a , x ] f ′ ( t ) d t + f ( a )

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Inherent Pitfall of Lebesgue Integration?I am studying Real Analysis with Royden&#039;s Book. I noticed that for a function   f differentiable almost everywhere on [a,b] and   f′ integrable over [a,b], it does not imply that   f  (  x  )  =      ∫          [      a      ,      x      ]            f    ′    (  t  )    d  t  +  f  (  a  )  .. Take Cantor function f as a counter example, since   1  =  f  (  1  )  &amp;gt;      ∫          [      0      ,      1      ]            f    ′    (  t  )    d  t  +  f  (  0  )  =  0  ,, because   ∀  x  ∈  [  0  ,  1  ]. However, does it mean that Lebesgue integral has some inherent pitfall in the definition of integration? In a sense that it fails to restore the original picture of a function in a set of measure zero after differentiation. Is that any better (or more general) design of integration? so that we can guarantee   f  (  x  )  =      ∫          [      a      ,      x      ]            f    ′    (  t  )    d  t  +  f  (  a  )

Illuddybopylu · Accepted Answer

When we get to Lebesgue integration, we typically do so because we have identified pathologies in which the (conventional) Riemann integration fails. It is a natural question to ask whether the Lebesgue integral belongs to a measure theoretic version of the Fundamental Theorem of Calculus. And while at first it may not appear to be a big deal, it doesn&#039;t really make sense to attempt to establish the fundamental theorem of calculus using measure-theoretic (Lebesgue) integration and conventional differentiation.Indeed, the standard way of defining a derivative uses limits of ratios of differences; differences make sense when we have well-defined sets and the notion of measure coincides with our notion of 1-dimensional absolute value. So it instead might make sense to explore not a different kind of integration, but a different kind of differentiation that works in a measure theoretic sense.This is in fact the essence of the Radon-Nikodym derivative. This derivative uses the fact that given a measurable set E and a measurable function f, the mapping   E  ↦      ∫    E    f    d  μ is a signed measure, and we denote this relationship with the notation   d  ν  =  f  d  μ; moreover, it is true that given   σ-finite measures   μ,   ν, with   ν positive and   ν=0 whenever   μ  =  0, that we can write  d  ν  =            d      ν              d      μ        d  μ  .We call             d      ν              d      μ       the Radon-Nikodym derivative. Note, this is actually a special case of a slightly more broad statement.Using a bit more care in restricting our domain, we can actually develop an analog to the Fundamental Theorem of Calculus, known as the Lebesgue Differentiation Theorem, which states that for a locally integrable function f, for a.e. x, we have       lim          r      →      0            1          m      (              E        r            )            ∫                  E        r              f  (  y  )    d  y  =  f  (  x  ) for every       E    r   that &quot;shrinks nicely&quot; to x.So we can do what you want to do, but we have to use a notion of the derivative that melds with what we do in a measure theoretic sense. You can&#039;t play basketball with hockey pucks, but you can sometimes play both in the same arena.

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