The version of the fundamental theorem of calculus found in Calculus says that if f is a
Riemann integrable function, the function
∫ x
x → f (t)dt,
a
has a derivative at every point where f is continuous. It is natural to ask what occurs for f in
L^{1}. It is an amazing fact that the same result is obtained aside from a set of measure zero
even though f, being only in L^{1} may fail to be continuous anywhere. Proofs of this result are
based on some form of the Vitali covering theorem presented above. In what follows, the
measure space is (ℝ^{n},S,m) where m is n-dimensional Lebesgue measure. To save notation, m
is written in place of m_{n}.
Also, to save on notation m is also the name of the outer measure defined on all of P(ℝ^{n})
which is determined by m_{n}. Also dx or dy will be used in place of dm or dm_{n} to emphasize
more familiar notation. Recall
B (p,r) = {x : |x − p| < r}. (17.1)
(17.1)
Also define the following.
If B = B (p, r), then ^B = B (p,5r). (17.2)
(17.2)
The first version of the Vitali covering theorem presented above will now be used
to establish the fundamental theorem of calculus. The space of locally integrable
functions is the most general one for which the maximal function defined below makes
sense.
Definition 17.1.1f ∈ L_{loc}^{1}(ℝ^{n}) means fX_{B(0,R)}∈ L^{1}(ℝ^{n}) for all R > 0. Forf ∈ L_{loc}^{1}(ℝ^{n}), the Hardy Littlewood Maximal Function, Mf, is defined by
| |
|| 1 ∫ ||
||m(B-(x,r))- f (y)dy − f(x )||
∫B (x,r)
≤ ----1----- |f(y)− f (x)|dy
m (B(x,r)) B(x,r)
and the last integral converges to 0 a.e. x.■
Definition 17.1.8 For N the set of Theorem 17.1.5or Corollary 17.1.6, N^{C}is called the Lebesgue setor the set of Lebesgue points.
The next corollary is a one dimensional version of what was just presented.
Corollary 17.1.9 Let f ∈ L^{1}(ℝ) and let
∫
F(x) = x f(t)dt.
− ∞
Then for a.e.x,F^{′}(x) = f(x).
Proof: For h > 0
∫ x+h ∫ x+h
1- |f(y)− f(x)|dy ≤ 2(-1-) |f(y) − f (x)|dy
h x 2h x−h
By Theorem 17.1.5, this converges to 0 a.e. Similarly
∫
1- x |f(y)− f(x)|dy
h x−h
converges to 0a.e.x.
|| || ∫ x+h
||F(x-+h-)−-F(x)− f(x)|| ≤ 1 |f(y)− f(x)|dy (17.4)
h h x
(17.4)
and
||F(x)− F (x− h) || 1∫ x
||------h-------− f(x)|| ≤ h |f(y)− f(x)|dy. (17.5)
x−h
(17.5)
Now the expression on the right in 17.4 and 17.5 converges to zero for a.e. x. Therefore, by
17.4, for a.e. x the derivative from the right exists and equals f
(x)
while from 17.5 the
derivative from the left exists and equals f