Measurable functions in measure theory are comparable to the continuous function in topology. In topology, continuous function maps an open set to an open set similarly, in measure theory, a measurable function maps a measurable set to another measurable set. A measurable space is a set A along with its non-empty collection of subsets of A, S, such that (A, S) satisfies two conditions:

• If A and B are among the collection of subsets in S, then A – B will be also a part of S.
• For any collection of subsets A₁, A₂, A₃, …, the union of all the A_i’s is equal to S.

The elements of the collection S are called measurable sets.

Thus, a measurable function is defined on a measurable space into a measurable space.

Definition of Measurable Functions

A function f: E→R* defined on a measurable set E into the set of extended real numbers R* is called Lebesgue measurable function or simply measurable function on E, if the set {x ∈ E| f(x)>a} is measurable for all real numbers a.

Measurable functions can be also defined as, let (A, X) and (B, Y) be measurable spaces and if f be a function from X into Y, that is, f: A→B is said to be measurable if f^-1(B) ∈ X for every B in Y.

With this, we have a very important theorem that is helpful for investigating the fact that whether a given function on a measurable set is measurable. The statement of the theorem is as follows:

These four conditions can be equivalently used to show a function being a measurable function.

Example of Measurable Function

A simple example of a measurable function could be the constant function defined on a measurable set.

Let f: E→R* be a constant function on a measurable set E, then we can define f as

f(x) = c where c is a constant. We can always find a real number ‘a’ such that c > a.

Then, {x ∈ E| f(x) > a} = E if c > a or {x ∈ E| f(x) > a} = Φ if c ≤ a.

By the above definition of measurable functions, both E and Φ are measurable sets.

Hence, f is a measurable function.

Properties of a Measurable Function

Following are some important and useful properties of a measurable function:

• If f is a measurable function defined on measurable sets E_n for all n being natural numbers, and if E = ⋃E_n, then f is measurable on E as well.
• If f is a measurable function on a measurable set A and B ⊂ A is a measurable set then f is measurable on B.
• If f is a continuous function defined on set E which is a measurable set, then f is a measurable function.
• A continuous function on a closed interval is measurable.
• A function f will be a measurable function on measurable set A, if and only if, for any open set G in R, f^-1(G) is a measurable set.
• If f and g are measurable functions, then f + g and fg are also measurable functions.
• If f is a measurable function on a measurable set E, and g is a continuous function defined on the range of f, then g is a measurable function on E.

Related Articles

• Continuity and Differentiability
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• Functions
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• Set Theory

Solved Examples on Measurable Functions

Example 1:

Show that the function f defined on R by

is a measurable function.

Solution:

Let R(f ≤ a) = { x ∈ R| f(x) ≤ a}, then

R(f ≤ a) = x + 5 ≤ a ⇒ x ≤ a – 5

Hence, we get

Obviously, each set is measurable. Hence, R(f ≤ a) is measurable for all real a. Thus f is a measurable function.

Example 2:

Let f be a function with a measurable domain D. Show that f is measurable if and only if the function g defined by g(x) = f(x) for x in D and g(x) = 0 for x not in D.

Solution:

As f is measurable then

E(g > a) = D(f > a) if a ≥ 0

= D(f > a) ∪ D^c if a < 0

Where D^c = complement of D

Since D is measurable ⇒ D^c is measurable

Since f is measurable ⇒ D(f > a) is measurable

⇒ D(f > a) ∪ D^c is measurable ⇒ E(g > a) is measurable

⇒ g is measurable function.

Now, we take g is measurable

again, E(g > a) = D(f > a) if a ≥ 0

= D(f > a) ∪ D^c if a < 0

Since E(g > a) is measurable ⇒ D(f > a) is measurable

⇒ f is a measurable function.