Measure, Integration & Real Analysis, 2021a

74 Chapter 3 Integration
3A
Integration with Respect to a Measure
Integration of Nonnegative Functions
We will first define the integral of a nonnegative function with respect to a measure.
Then by writing a real-valued function as the difference of two nonnegative functions,
we will define the integral of a real-valued function with respect to a measure. We
begin this process with the following definition.
3.1 Definition S-partition
Suppose S is a σ-algebra on a set X. AnS-partition of X is a finite collection
A 1 ,...,A m of disjoint sets in S such that A 1 ∪···∪A m = X.
The next definition should remind you
We adopt the convention that 0 · ∞
of the definition of the lower Riemann
and ∞ · 0 should both be interpreted
sum (see 1.3). However, now we are
to be 0.
working with an arbitrary measure and
thus X need not be a subset of R. More importantly, even in the case when X is a
closed interval [a, b] in R and μ is Lebesgue measure on the Borel subsets of [a, b],
the sets A 1 ,...,A m in the definition below do not need to be subintervals of [a, b] as
they do for the lower Riemann sum—they need only be Borel sets.
3.2 Definition lower Lebesgue sum
Suppose (X, S, μ) is a measure space, f : X → [0, ∞] is an S-measurable
function, and P is an S-partition A 1 ,...,A m of X. The lower Lebesgue sum
L( f , P) is defined by
m
L( f , P) = ∑ μ(A j ) inf f .
j=1
A j
Suppose (X, S, μ) is a measure space. We will denote the integral of an S-
measurable function f with respect to μ by ∫ f dμ. Our basic requirements for
an integral are that we want ∫ χ E
dμ to equal μ(E) for all E ∈S, and we want
∫ ( f + g) dμ =
∫ f dμ +
∫ g dμ. As we will see, the following definition satisfies
both of those requirements (although this is not obvious). Think about why the
following definition is reasonable in terms of the integral equaling the area under the
graph of the function (in the special case of Lebesgue measure on an interval of R).
3.3 Definition integral of a nonnegative function
Suppose (X, S, μ) is a measure space and f : X → [0, ∞] is an S-measurable
function. The integral of f with respect to μ, denoted ∫ f dμ, is defined by
∫
f dμ = sup{L( f , P) : P is an S-partition of X}.
Measure, Integration & Real Analysis, by Sheldon Axler