- #1
justin_huang
- 13
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If f : Rn -> R is Lebesgue measurable on Rn, prove that the function F : Rn * Rn -> R de fined by F(x, y) = f(x - y) is Lebesgue measurable on Rn * Rn.
how can I prove this question?
how can I prove this question?
ebola1717 said:You should provide us some detail about what attempts you've made. From what you've given, I'm not sure what you're stuck on and how I can help. I'll assume you've tried the obvious thing, which is look at the definition, F is measurable if [tex] F^{-1}(a,\infty][/tex] is measurable. Since f is measurable, we know [tex] f^{-1}(a,\infty][/tex] is measurable. Now try to relate the points in this set to the points in the former set. It might help to look at it visually. Now, using this process will get you thinking about the function in the right way, but there are easier ways to prove it. You might want to think about composing measurable functions.
A Lebesgue measurable function is a function that maps a measurable subset of its domain to a measurable subset of its range. In other words, the pre-image of any measurable set in the range must be a measurable set in the domain.
Unlike Riemann integrable functions, Lebesgue measurable functions do not require the existence of a limit for the upper and lower sums to be equal. Instead, they are defined using the Lebesgue measure, which allows for a more general definition of integration that does not rely on limits.
Proving a function is Lebesgue measurable is important because it allows for the use of more powerful integration techniques, such as the Lebesgue integral, which can handle a wider range of functions than the Riemann integral. It also allows for the development of more general and abstract theories in areas such as measure theory and functional analysis.
The Lebesgue measure is used to define the Lebesgue integral, which is used in the definition of a Lebesgue measurable function. The measure assigns a value to each measurable set, which is then used to calculate the integral of a function over that set. This allows for a more flexible and general definition of integration.
Yes, a function can be Lebesgue measurable without being continuous. The Lebesgue measure is a more general and flexible concept than continuity, so a function can be measurable even if it has discontinuities or is undefined at certain points. However, continuous functions are always Lebesgue measurable.