Prove f>=0 almost everywhere

  • Thread starter jinsing
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In summary: That makes sense, I'll add that in.In summary, if {f_n} is a collection of measurable functions on R satisfying the given criteria, then f(x) >= 0 almost everywhere on R, where f(x) = inf{f_n(x)|n in N}. This can be proven by first defining S_n = {x | f_n(x) < 0} and showing that the measure of S_n is 0 for all n in N. Then, by taking the union of all S_n, it can be shown that f(x) >= 0 for all x in R-S, where S is the union of all S_n with measure 0. Therefore, f(x) >= 0 almost everywhere
  • #1
jinsing
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Homework Statement


If {f_n} is a collection of measurable functions defined on R and satisfying: |f_n(x)|<= 1 for all n in N and x in R and f_n(x) >=0 almost everywhere on R for all n in N and f(x) = inf{f_n(x)|n in N}, then f(x) >= 0 almost everywhere on R.


Homework Equations



Almost everywhere, sequences of functions, measurability


The Attempt at a Solution



Assume {f_n} is a collection of measurable functions defined on R satisfying the criteria above. Since for all n in N f_n(x) >= 0 a.e on R, define S_n = {x | f_n(x) < 0}. Then by definition for all n in N m(S_n) = 0. Now let [tex] S = \bigcup_{n=1}^\infty S_n. [/tex] Since S is the union of several sets S_n with measure of zero, then m(S) = 0. So, we know f(x) = inf{f_n(x) | n in N} >= 0 for all x in R-S. But m(S) = 0, so f(x) >= 0 almost everywhere, as desired.

I'm wondering if I can necessarily state that the inf of the f_n's will be >= 0. I'm not that great at catching little details like that, so having someone give the proof a quick look-over would be incredibly helpful.

Thanks!
 
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  • #2
It helps to clarify things if you fix x to be a specific value, say b. If [itex]f_n(b) \geq [/itex] for all values of n, then is [itex] \inf\{f_n(b)} \geq 0[/itex]? This follows immediately from the definition of infimum.

Then conclude that S can only consist of values of x for which at least one fn(x)<0
 
  • #3
Ah! Thanks for the input!
 

What does it mean to "prove f>=0 almost everywhere"?

Proving f>=0 almost everywhere means showing that the function f is non-negative except for a set of points with measure zero. In other words, f is equal to or greater than 0 at almost every point in its domain.

Why is it important to prove f>=0 almost everywhere?

Proving f>=0 almost everywhere is important because it is a crucial step in verifying the positivity of a function, which is often necessary in mathematical and scientific applications. It also helps ensure the validity and accuracy of results obtained using the function.

What techniques are commonly used to prove f>=0 almost everywhere?

There are several techniques that can be used to prove f>=0 almost everywhere, depending on the specific function and context. These include the use of measure theory, integration methods, and functional analysis techniques.

Can f be negative at any point if f>=0 almost everywhere?

Technically, yes, f can be negative at a finite number of points and still satisfy the condition of f>=0 almost everywhere. However, this is typically a rare occurrence and is usually excluded when discussing the positivity of a function.

Can f>=0 almost everywhere be proven for all functions?

No, f>=0 almost everywhere cannot be proven for all functions. There are certain functions, such as those with oscillating behavior or those that do not have a well-defined domain, for which this condition may not hold. It is important to carefully consider the properties and limitations of the specific function being analyzed when attempting to prove f>=0 almost everywhere.

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