Is the Product of Integrable Functions also Integrable?

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In summary, integrable functions are those that can be represented by a definite integral, which is a mathematical concept that represents the area under a curve. The product of two integrable functions is always integrable, and it has various practical applications in fields such as physics, engineering, and economics. However, there are limitations to the concept of integrable functions, as certain types of functions cannot be represented by a definite integral and are considered non-integrable.
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Chris L T521
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Here's this week's problem!

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Problem: Let $h$ and $g$ be integrable functions on $X$ and $Y$, and define $f(x,y)=h(x)g(y)$. Show that if $f$ is integrable on $X\times Y$ with respect to the product measure, then
\[\int_{X\times Y} f\,d(\mu\times\nu) = \int_X h\,d\mu \int_Y g \,d\nu.\]

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Note: We do not need to assume that $\mu$ and $\nu$ are $\sigma$-finite.

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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No one answered this week's problem. I don't have a solution ready at this time; it's on my backlog of solutions to write and I'll try to get them done soon. I take the GRE on Saturday so it's been crunch time for me. Thank you for your patience!
 

FAQ: Is the Product of Integrable Functions also Integrable?

What does it mean for a function to be integrable?

Integrable functions are those that can be represented by a definite integral, which is a mathematical concept that represents the area under a curve. In simpler terms, an integrable function is a function that can be measured and compared in terms of area.

Is the product of two integrable functions always integrable?

Yes, the product of two integrable functions is also integrable. This can be proved using the properties of integrals and the fact that the product of two continuous functions is also continuous.

Can the product of two integrable functions be non-integrable?

No, the product of two integrable functions cannot be non-integrable. This is because if both functions are integrable, then their product will also be integrable. However, the converse is not always true - a non-integrable function can have a product that is integrable.

What is the significance of considering integrable functions?

Integrable functions have various practical applications in fields such as physics, engineering, and economics. They allow us to calculate important quantities such as areas, volumes, and probabilities. They also play a crucial role in the fundamental theorem of calculus.

Are there any limitations to the concept of integrable functions?

Yes, there are certain types of functions that are not integrable, such as functions that have infinite discontinuities or oscillate infinitely. These types of functions cannot be represented by a definite integral and are considered non-integrable.

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