So many Integrals – II

Standard

As promised in previous post, now I will briefly discuss the remaining two flavors of Integrals.

Stieltjes Integral

ss

Stieltjes

In 1894, a Dutch mathematician, Thomas Stieltjes, while solving the moment problem, that is, given the moments of all orders of a body, find the distribution of its mass, gave a generalization of the Darboux integral.

Let P : a = x_0 < x_1 < x_2<\ldots < x_n = b, n being an integer, be a partition of the interval [a, b].

For a function \alpha, monotonically increasing on [a,b], we write:

\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1})

Let f be a bounded function defined on an interval [a, b],\quad a, b being real numbers. We define the sum 

S_P = \sum_{i=1}^n f(t_i)\Delta \alpha_i, \quad \overline{S}_P = \sum_{i=1}^n f(s_i)\Delta \alpha_i

where t_i,s_i \in [x_{i-1} , x_i] be such that

f(t_i) = \text{sup} \{ f(x) : x \in [x_{i-1}, x_{i}]\},

f(s_i) = \text{inf} \{ f(x) : x \in [x_{i-1}, x_{i}]\}

If the \text{inf}\{S_P\} and \text{sup}\{\overline{S}_P\} are equal, we denote the common value by  \int_{a}^{b} f(x) d\alpha(x) and call it Steiltjes integral of f with respect to \alpha over [a,b].


 

Lebesgue Integral

leb.jpg

Lebesgue

Let me quote Wikipedia article:

The integral of a function f between limits a and b can be interpreted as the area under the graph of f. This is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions? In general, for which class of functions does “area under the curve” make sense? The answer to this question has great theoretical and practical importance.

In 1901, a French mathematician, Henri Léon Lebesgue generalized the notion of the integral by extending the concept of the area below a curve to include functions with uncountable discontinuities .

Lebesgue defined his integral by partitioning the range of a function and summing up sets of x-coordinates belonging to given y-coordinates, rather than, as had traditionally been done, partitioning the domain.

Lebesgue himself, according to his colleague, Paul Montel, compared his method with paying off a debt: (see:pp. 803,  The Princeton Companion to Mathematics)

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.

A set \mathcal{A} is said to be Lebesgue measurable, if for each set \mathcal{E} \subset \mathbb{R} the Carathéodory condition:

m^{*} (\mathcal{E}) = m^{*}(\mathcal{E} \cap \mathcal{A}) + m^{*}(\mathcal{E}\backslash \mathcal{A})

is satisfied, where m^{*}(\mathcal{A}) is called outer measure and is defined as:

m^{*}(\mathcal{A}) = \inf\sum\limits_{n=1}^\infty (b_n-a_n)

where \mathcal{A} is a countable collection of closed intervals [a_n,b_n], a_n\leq b_n, that cover \mathcal{A}.

The Lebesgue integral of a simple function \phi(x) = \sum_{i=1}^n c_i \chi_{\mathcal{A}_i} (x) on \mathcal{A}, where \mathcal{A}=\bigcup_{i=1}^{\infty} \mathcal{A}_{i}, \mathcal{A}_i are pairwise disjoint measurable sets and c_1, c_2, \ldots are real numbers, is defined as:

\int\limits_{\mathcal{A}} \phi dm = \sum\limits_{i=1}^{n} c_i m(\mathcal{A}_i)

where, m(\mathcal{A}_i) is the Lebesgue measure of a measurable set \mathcal{A}_i.

An extended real value function f: \mathcal{A}\rightarrow \overline{\mathbb{R}} defined on a measurable set \mathcal{A}\subset\mathbb{R} is said to be Lebesgue measurable on \mathcal{A} if f^{-1} ((c,\infty]) = \{x \in\mathcal{A} : f(x) > c\} is a Lebesgue measurable subset of \mathcal{A} for every real number c.

If f is Lebesgue measurable and non-negative on \mathcal{A} we define:

\int\limits_{\mathcal{A}} f dm = \sup \int\limits_{\mathcal{A}} \phi dm

where the supremum is taken over all simple functions \phi such that 0\leq \phi \leq f.

The function f is said to be Lebesgue integrable on \mathcal{A} if it’s integral over \mathcal{A} is finite.


The Lebesgue integral is deficient in one respect. The Riemann integral generalizes to the improper Riemann integral to measure functions whose domain of definition is not a closed interval. The Lebesgue integral integrates many of these functions, but not all of them.

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2 responses »

  1. Pingback: Integration & Summation | Gaurish4Math

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