Lebesgue Integration - Basic Theorems of The Lebesgue Integral

Basic Theorems of The Lebesgue Integral

The Lebesgue integral does not distinguish between functions which differ only on a set of μ-measure zero. To make this precise, functions f and g are said to be equal almost everywhere (a.e.) if

• If f, g are non-negative measurable functions (possibly assuming the value +∞) such that f = g almost everywhere, then

To wit, the integral respects the equivalence relation of almost-everywhere equality.

• If f, g are functions such that f = g almost everywhere, then f is Lebesgue integrable if and only if g is Lebesgue integrable and the integrals of f and g are the same.

The Lebesgue integral has the following properties:

Linearity: If f and g are Lebesgue integrable functions and a and b are real numbers, then af + bg is Lebesgue integrable and

Monotonicity: If f ≤ g, then

Monotone convergence theorem: Suppose {f_k}_{k ∈ N} is a sequence of non-negative measurable functions such that

Then, the pointwise limit f of f_k is Lebesgue integrable and

Note: The value of any of the integrals is allowed to be infinite.

Fatou's lemma: If {f_k}_{k ∈ N} is a sequence of non-negative measurable functions, then

Again, the value of any of the integrals may be infinite.

Dominated convergence theorem: Suppose {f_k}_{k ∈ N} is a sequence of complex measurable functions with pointwise limit f, and there is a Lebesgue integrable function g (i.e., g belongs to the space L1) such that |f_k| ≤ g for all k.

Then, f is Lebesgue integrable and