When dealing with double Lebesgue integrals, just as with...

When dealing with double Lebesgue integrals, just as with double Riemann integrals, the order of integration can be reversed. The only 1.9 Exercises 43 assumption required is that the function being integrated be either nonnegative or integrable. Here is an application of this fact. Let $X$ be a nonnegative random variable with cumulative distribution function $F(x)=\mathbb{P}\{X \leq x\}$. Show that $$ \mathbb{E} X=\int_0^{\infty}(1-F(x)) d x $$ by showing that $$ \int_{\Omega} \int_0^{\infty} \mathbb{I}_{[0, X(\omega))}(x) d x d \mathbb{P}(\omega) $$ is equal to both $\mathbf{E} X$ and $\int_0^{\infty}(1-F(x)) d x$.

When dealing with double Lebesgue integrals, just as with double Riemann integrals, the order of integration can be reversed. The only
1.9 Exercises
43
assumption required is that the function being integrated be either nonnegative or integrable. Here is an application of this fact.

Let $X$ be a nonnegative random variable with cumulative distribution function $F(x)=\mathbb{P}\{X \leq x\}$. Show that
$$
\mathbb{E} X=\int_0^{\infty}(1-F(x)) d x
$$
by showing that
$$
\int_{\Omega} \int_0^{\infty} \mathbb{I}_{[0, X(\omega))}(x) d x d \mathbb{P}(\omega)
$$
is equal to both $\mathbf{E} X$ and $\int_0^{\infty}(1-F(x)) d x$.

Key Concepts

Double Lebesgue Integration

Double Lebesgue integration involves computing the integral of a function of two variables over a product space using the Lebesgue integral. It is foundational in advanced calculus and measure theory, allowing integration over sets where the function or domain may have complex structure, and it is particularly useful when functions cannot be handled using elementary Riemann integration techniques.

Fubini's and Tonelli's Theorems

Fubini's theorem provides conditions under which the order of integration in a double integral can be reversed, while Tonelli's theorem extends this idea to nonnegative functions. These theorems are crucial in measure theory because they justify changing the order of integration, a technique often employed in simplifying complex integrals or proving identities in probability theory and analysis.

Indicator Functions

Indicator functions are simple functions that take the value 1 on a specified set and 0 outside it. They are used to 'select' parts of a domain within an integral, and are essential tools in measure theory and probability to represent events and to simplify the expressions of integrals by converting them into integrals over measurable sets.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable describes the probability that the variable takes on a value less than or equal to a given number. In probability theory, the CDF provides a complete description of the probability distribution of a random variable, and it is often linked to integrals that represent probabilities and expectations.

Expectation as an Integral

The expectation of a nonnegative random variable can be expressed as an integral over its probability distribution, often formulated using indicator functions and the CDF. This integral representation connects measure theory and probability, illustrating how integral calculus is used to compute the average or expected value of a random variable over its entire range.

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