The Lebesgue integrals are the integration of functions over measurable sets, which could integrate many functions that cannot be integrated as Riemann integrals or even Riemann-Stieltjes integrals. The concept behind the Lebesgue integrals is that generally, while integrating a given function, the total area under the curve is divided into several vertical rectangles, but while determining the Lebesgue integral of the function, the area under the curve is divided into horizontal slabs, that need not be rectangles.

Also, learn about the Riemann Integrals.

In Lebesgue Integral, rather than partitioning the domain of the given function, the range of the function is partitioned. Before defining the Lebesgue integrals, we shall define the simple functions.

Simple Function: A linear combination

is called a simple function, where the sets E_i = { x | 𝜙(x) = a_i} are disjoint and measurable, the numbers a_i’s are non-zero and distinct, 𝝌_Ei is the characteristic function of E_i. That is, for x ∈ E_i, 𝝌_Ei = 1, and if x not in E_i ⇒ 𝝌_Ei = 0.

Integral of Simple Function: If 𝜙 is a simple function that vanishes outside a set of finite measure E_i, its integral is defined as

where m(E_i) is the measure of set E_i, (i = 1, 2, 3, …, n).

Definition of Lebesgue Integral

Let f: E → R be a bounded function and E be a measurable set of finite measure. Then we may have the numbers

inf _{𝜓 ≥ f} ∫_E f(x) dx and sup_{𝜙 ≤ f} ∫_E f(x) dx,

where 𝜓 and 𝜙 are simple functions over measurable set E. These two numbers exist and are respectively called upper Lebesgue integral and lower Lebesgue integral.

L⨜_E f(x) dx = inf _{𝜓 ≥ f} ∫_E f(x) dx

The upper Lebesgue integral

L⨜_E f(x) dx = sup_{𝜙 ≤ f} ∫_E f(x) dx

The lower Lebesgue integral.

Lebesgue Integrable Function

Let f be a simple bounded function defined on a set E of finite measure is said to be Lebesgue integrable over E, if

L⨜_E f(x) dx = L ∫_E f(x) dx =L⨛_E f(x) dx

and the common value L ∫_E f(x) dx or simply, ∫_E f(x) dx of both upper and lower Lebesgue integral is called the Lebesgue Integral of f.

Properties of Lebesgue Integral

Let f and g be Lebesgue integrable; that is, bounded measurable functions defined on a measurable set E of finite measure. Then

• ∫_E (a f) = a ∫_E f ∀ a ∈ R
• ∫_E (f ± g)= ∫_E f ± ∫_E g
• ∫_E (a f ± b g)= a ∫_E f + b ∫_E g where a, b ∈ R
• If f = g almost everywhere, then ∫_E f = ∫_E g
• If f ≤ g almost everywhere, then ∫_E f ≤ ∫_E g
• |∫_E f| ≤ ∫_E | f |
• If 𝛼 ≤ f(x) ≤ 𝛽, then 𝛼 m(E) ≤ ∫_E f(x) dx ≤ 𝛽 m(E). This result is known as the First Mean Value Theorem.
• If A and B are disjoint measurable subsets of E, then ∫ _{A ∪ B} f = ∫ _A f + ∫ _B f.
• If f(x) ≥ 0 on E, then ∫_E f(x) dx ≥ 0; and if f(x) ≤ 0 on E, then ∫_E f(x) dx ≤ 0.
• If m(E) = 0, then ∫_E f = 0.
• If f(x) = k almost everywhere on E, then ∫_E f = k m(E). In particular, if f(x) = 0, then ∫_E f = 0 and if f(x) = 1, then ∫_E f = m(E).

Related Articles

• Measurable Functions
• Fatou’s Lemma
• Sequence and Series of Functions
• Power Series

Lebesgue Integral of Non-negative Measurable Functions

Let f be a non-negative measurable function defined on a measurable set E of finite measure. Then the Lebesgue integral of f over E is defined by

∫_E f = sup _{h ≤ f} ∫_E h

where h is a bounded measurable function over E such that m({x ∈ E | h(x) ≠ 0}) is finite.

Lebesgue Integral for Unbounded Function

Let f be a non-negative measurable function over measurable set E of finite measure. For each x ∈ E and n ∈ N, let us define a function f_n(x) as:

If lim_{n → ∞} ∫_E f_n(x) dx exists finitely, we say that the unbounded function f is Lebesgue Integrable and ∫_E f = lim_{n → ∞} ∫_E f_n(x) dx.

Solved Example of Lebesgue Integrals

Example :

Evaluate the Lebesgue Integral of the function f: [0,1] → R defined by

and show that f is Lebesgue integrable on [0, 1].

Solution:

Since 1/x^3/2 → ∞ as x → 0, so f is unbounded in [0, 1]. To prove that f is Lebesgue integrable, define

Now,

Thus, by the definition of Lebesgue integral of an unbounded function,

Hence, Lebesgue integral of f over [0, 1] is 3/2, and f is Lebesgue integrable over [0, 1].