Help with a real analysis problem

In summary, real analysis is a branch of mathematics that deals with the rigorous study of real numbers and their properties. Some common problems in real analysis include proving convergence or divergence of sequences and series, finding extrema of functions, and evaluating integrals. To improve problem-solving skills in real analysis, it is important to have a strong understanding of fundamental concepts and practice solving problems. Common mistakes to avoid include not fully understanding the problem or theorems, not showing all steps in a proof, and making computational errors. Real analysis also has applications in other fields such as physics, engineering, and economics.
  • #1
polkadot66
1
0
Homework Statement
Given ## f \in L^1(X, M, \mu)## , show there is a ## \sigma## -finite measure ## \mu'## such that ## \int_{E} f d\mu = \int_{E} f d\mu'## .
Relevant Equations
## f \in L^1## so ## \int |f| d\mu < \infty##
a measure ## \mu'## is ## \sigma##-finite if there are sets ##A_1, A_2, ...## such that ## \mu'(A_n) < \infty## and ##X= \cup A_n##
I tried to prove this by absurd stating that there is no such ## \mu'## but i couldn't get anywhere...
 
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  • #2
i would begin with the definition of f being in L^1. That should give you a sequence of step functions that approximate f and can be used to define some sets of finite measure that can be used to modify mu. anyway i would do it directly, not by contradiction.
 

Related to Help with a real analysis problem

1. What is real analysis?

Real analysis is a branch of mathematics that deals with the study of real numbers and their properties. It involves the use of mathematical techniques to understand and analyze real-valued functions, sequences, and series.

2. Why is real analysis important?

Real analysis is important because it provides a foundation for many other branches of mathematics, such as calculus and differential equations. It also has numerous applications in physics, engineering, and economics.

3. What are some common topics in real analysis?

Some common topics in real analysis include limits, continuity, differentiation, integration, and sequences and series. Other topics may include metric spaces, topological spaces, and measure theory.

4. What are some useful techniques for solving real analysis problems?

Some useful techniques for solving real analysis problems include the use of mathematical theorems and proofs, understanding and applying definitions, and using algebraic and trigonometric manipulations. Visual aids, such as graphs and diagrams, can also be helpful in understanding and solving problems.

5. How can I improve my skills in real analysis?

To improve your skills in real analysis, it is important to practice solving a variety of problems and understanding the underlying concepts. Reading textbooks and attending lectures or online courses can also be helpful in gaining a deeper understanding of the subject. Additionally, seeking guidance and feedback from a mentor or tutor can aid in improving your skills in real analysis.

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