Purchase Solution

Real Analysis : Measurable Sets and Functions

Not what you're looking for?

Ask Custom Question

1).If f: X--> C ( C is complex plane) is measurable, then prove that f^-1({0}) ( f inverse of 0 or any other point) is a measurable set in X.

2). If E is measurable set in X and if

X_E ( x) = { 1 if x is in E, 0 if x is not in E} then X_E is a measurable function. Now I want you to prove the other direction, that is, I want you to show that
If X_E(x) is measurable, then E is measurable.
I believe that X_E(x) here is the characteristic function ( not sure about the name, but it is defined above).

Purchase this Solution
Solution Summary

Measurable sets and functions are investigated. The characteristic functions is analyzed.

Solution Preview

1. Proof:
Since f: X-->C is measurable, then for any open set A in C, f^(-1)(A) is a measurable set. We note that A=C-{0} is an open set, thus ...

Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Probability Quiz

Some questions on probability

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

View More Free Quizzes
Related BrainMass Solutions

  • Real Analysis : Lebesgue Measure

    38795 Real Analysis : Lebesgue Measure Show that the sum and product of two simple functions are simple.

  • Real analysis proofs

    Since all are measurable, then are measurable sets. Hence is measurable. Therefore, is a measurable function. There are two real analysis proofs here, one regarding unions and supremum and one regarding measurable functions.

  • Measurable Spaces and Properties of Integrals of Simple Functions

    Measurable Sets and Spaces and Properties of Integrals of Simple Functions are investigated. The solution is detailed and well presented.

  • Counting measure

    (Generally counting measures is applied on sets like N and Z) It is true that to construct a measurable function, you can always start from simple functions{ai's} Such that each ai = (1) Where denotes the characteristics function whose

  • Measurable function

    Proof First, if , that is, is the point-wise limit of simple functions, then check that , then is measurable. Now let be a measurable function. We fix and define , These sets are measurable. Now let .

  • Prove Functions and Sets

    46695 Prove Functions and Sets (See attached file for full problem description) 1) a.

  • Real analysis : Measurable Functions

    24480 Real analysis : Measurable Functions Q1.

  • Integrals of measurable functions

    55356 Integrals of measurable functions Let X be an uncountable set, let m be the collection of all sets E in X such that either E or E^c is at most countable, and define M(E) = 0 in the first case, and M(E) = 1 in the second case. ( m here is sigma algebra

  • Lebesgue Measurable Sets, Compact Sets and Open Sets

    Mathematics, Real Variables lebesgue sets and compact sets problem If A is lebesgue measurable sets in R^n, bounded, then there is a compact set K_epsilon and an open set for every epsilon > 0 V_epsilon such that K_epsilon is subset of A

View More Related Solutions