Theory of Statistics - George Mason University

0.1 Measure, Integration, and Functional Analysis 721
in which we adopt the conventions above so that if, in any addend in expression
(0.1.39), either factor is 0, then the addend is 0. The definition in
equation (0.1.38) is therefore equivalent to
f dν = sup
all partitions
i
inf
ω∈Ai
f(ω)ν(Ai) (0.1.40)
and so again the integral (0.1.38) is always defined, although it may be infinite.
Now consider general Borel functions. For a general Borel function f, we
form two nonnegative Borel functions f+ and f− such that f = f+ − f−:
f+(ω) = max{f(ω), 0}
f−(ω) = max{−f(ω), 0}.
Definition 0.1.39 (integral of a general Borel function)
The integral of f with respect to ν is the difference of the integrals of the two
nonnegative functions:
f dν =
f+ dν −
f− dν, (0.1.41)
so long as either f+ dν or f− dν is finite (because ∞ − ∞ is not defined).
We can rewrite the definition in equation (0.1.41) in a manner similar to
how we rewrote equation (0.1.39) above:
f dν = sup
all partitions
inf
f(ω)
ω∈Ai
ν(Ai). (0.1.42)
Note that, just as with the definitions for nonnegative functions above, the
integral of a general Borel function may be infinite; in fact, it may be ∞ or
−∞.
For what kind of function would the Lebesgue integral not be defined?
The Lebesgue integral is not defined for functions for which both the positive
part and the negative of the negative part in equation (0.1.41) are ∞. The
function f(x) = sin(x)/x over the positive real line is an example of such a
function (but see the section beginning on page 730).
Although the definition allows the integral to be infinite, we use a special
term for the case in which the integral is finite. If both f+ dν and f− dν are
finite, the integral itself is finite, and in that case we say that f is integrable.
Note that being Borel does not imply that a function is integrable.
We define the integral over a domain A as
f dν = IAf dν. (0.1.43)
A
Although we may not explicitly identify the underlying measure space, technically
there is one, say (Ω, F, ν), and A ∈ F and so A ⊆ Ω.
Theory of Statistics c○2000–2013 James E. Gentle
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