Not quite, there are two things wrong here. First, in a sigma-algebra, you must use countable unions and intersections. So instead of $t$ you should have $1/n$. Then you can take the intersection of the sets $\bigl(a,b+\frac1n\bigr]$. That leads us to the second thing that is wrong, namely the fact that \(\displaystyle \bigcap_n \bigl(a,b+\tfrac1n\bigr] = (a,b]\): you still have an interval that is closed at the upper end!Fermat said:I am trying to show that an open set in [0,1] is measurable, given that [0,x] is measurable set for each x in [0,1]. So I need to show (a,b) is measurable. Using the fact that measurable sets form a sigma algebra, I have managed to show that (a,b] is measurable. So (a,b+t] is measurable for any t>0. letting t ->0, can I then conclude that (a,b) is measurable?
The "measurability" of open sets in [0,1] refers to whether or not these sets can be measured or assigned a numerical value in terms of their size or volume. In mathematical terms, it is a measure of the extent to which these sets can be quantified.
The measurability of open sets is determined by whether or not they satisfy certain mathematical criteria. In particular, a set is considered measurable if it can be approximated by other sets with known measures.
The measurability of open sets is important in mathematics and science because it allows us to quantify and analyze these sets, which can help us understand and make predictions about various phenomena. It is also a fundamental concept in measure theory, which has numerous applications in fields such as physics, economics, and engineering.
No, not all open sets in [0,1] are measurable. In fact, the set of non-measurable open sets in [0,1] is large and infinite. This is because, in general, it is difficult to define a measure for these sets in a consistent and meaningful way.
The measurability of open sets in [0,1] is closely related to other mathematical concepts such as Lebesgue measure, which is a way of assigning a numerical value to a set in [0,1] based on its size or volume. It is also related to the concepts of measure spaces and sigma-algebras, which are used to define and study measurable sets in a more general setting.