If the pre-image of the function is whole real line and is defined as following: f ( x

Lucia Grimes 2022-07-05 Answered
If the pre-image of the function is whole real line and is defined as following:
f ( x ) = { 1 if x Z 0 otherwise
What would be the essential supremum?
I understand that the essential supremum of a function is the smallest value that is larger or equal than the function values almost everywhere when allowing for ignoring what the function does at a set of points of measure zero.
Would it be still zero considering each individual integer essentially has measure zero? However, the measure of the integer set with value 1 is not zero, is it?
Thanks in advance.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Brendan Bush
Answered 2022-07-06 Author has 14 answers
Yes, indeed the essential supremum of this function is zero given the fact that union of sets of measure zero is still zero.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-06-17
State the number of possible triangles that can be formed using the given measurements, then sketch and solve the triangles, if possible.
m A = 48 , c = 29 , a = 4
asked 2022-06-11
With the help of a suitable transformation and Fubini I want to determine the integral
V x 3 y d λ 2 ( x , y ) ,
where V is the open subset of R + 2 bounded by the following curves:
x 2 + y 2 = 4 x 2 y 2 = 2 x 2 y 2 = 1
I know how to do that. The only problem is finding V . Is it
V = { ( x , y ) R 2 : 1 < x 2 y 2 < 2 , 0 < x 2 + y 2 < 4 }
Because then I set
x 2 + y 2 = v x 2 y 2 = u
and get
( x , y ) = ( 1 / 2 ( v u ) , 1 / 2 ( u + v ) )
So either my transformation is wrong or the limits of the intervalls I chose.
Thanks for any kind of help.
asked 2022-05-22
The equation d = | A x + B y + C z + D | ( A 2 + B 2 + C 2 ) gives is the distance between a plane and a point as the value 'd'. What is the unit of this value? Why is the distance between the plane and the point not a vector?
asked 2022-07-03
Three distinct positive integers are selected at random from 9 consecutive positive integers. What is the probability that their average is also an integer?
Is there any way to find it efficiently? I'd taken example of 1,2,3,4,5,6,7,8,9 then there are a lot of combinations to give a number divisible by three which give an integer average value. It's difficult to find the way :D
Thank you.
asked 2022-05-20
Let f be a nonegative measurable function on R . Show that
lim n n n f = R f
Of course
n , n n f R f
To show the reverse, I feel that I need to use Fatou's lemma; but can't seem to get the right argument.
asked 2022-04-13
For some measuring processes, the uncertainty is approximately proportional to the value of the measurement. For example, a certain scale is said to have an uncertainty of ±2%. An object is weighed on this scale.
a) Given that the reading is 100 g, express the uncertainty in this measurement in grams.
b) Given that the reading is 50 g, express the uncertainty in this measurement in grams.
asked 2022-06-14
Let μ be a complex measure on R n and f L 1 ( μ ) such that f ( x ) c > 0 ( for some constant c > 0 ) a.e. x R n . Then is it true that
| R n f d μ | | R n c d μ | = c | μ ( R n ) |  ? ?

My first thinking process is to break each integral first in its real and imaginary part and then their corresponding positive and negative parts, i.e.
R n f d μ = R e ( R n f d μ ) + i I m ( R n f d μ )
= R e + ( R n f d μ ) R e ( R n f d μ ) + i I m + ( R n f d μ ) i I m ( R n f d μ )
and then wanted to look at corresponding decompositions of the measure and estimate on each such segements. Also note that a complex measure by definition gives | μ ( R n ) | < . Now how to proceed from here.
Can someone please help?