Measure Theory/Integration

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Let be a -finite measure space. Suppose is a positive simple measurable function, with ; are disjoint.


Let be measurable, and let .


Now let be any measurable function. We say that is integrable if and are integrable and if . Then, we write

The class of measurable functions on is denoted by

For , we define to be the collection of all measurable functions such that

A property is said to hold almost everywhere if the set of all points where the property does not hold has measure zero.


Let be a measure space and let be measurable on . Then

  1. If , then
  2. If , , then
  3. If and then
  4. If , , then , even if
  5. If , , then , even if


Monotone Convergence Theorem[edit]

Suppose and are measurable for all such that

  1. for every
  2. almost everywhere on



is an increasing sequence in , and hence, (say). We know that is measurable and that . That is,



Define ; . Observe that and

Suppose . If then implying that . If , then there exists such that and hence, .

Thus, , therefore . As this is true if , we have that . Thus, .

Fatou's Lemma[edit]

Let be measurable functions. Then,


For define . Observe that are measurable and increasing for all .

As , . By monotone convergence theorem,

and as , we have the result.

Dominated convergence theorem[edit]

Let be a complex measure space. Let be a sequence of complex measurable functions that converge pointwise to ; , with

Suppose for some then

and as


We know that and hence , that is,

Therefore, by Fatou's lemma,

As , implying that


  1. Suppose is measurable, with such that . Then almost everywhere
  2. Let and let for every . Then, almost everywhere on
  3. Let and then there exists constant such that almost everywhere on


  1. For each define . Observe that
    but Thus for all , by continuity, almost everywhere on
  2. Write , where are non-negative real measurable.
    Further as are both non-negative, each of them is zero. Thus, by applying part I, we have that vanish almost everywhere on . We can similarly show that vanish almost everywhere on .