# Outer measure ... Axler, Result 2.14 ... Another Question ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...

I need further help with the proof of Result 2.14 ...

Result 2.14 and its proof read as follows

In the above proof by Axler we read the following:

" ... ... To get started with the induction, note that 2.15 clearly implies 2.16 if $$\displaystyle n = 1$$ ... "

Can someone please demonstrate rigorously that 2.15 clearly implies 2.16 if $$\displaystyle n = 1$$ ...

... in other words, demonstrate rigorously that $$\displaystyle [a, b] \subset I_1 \Longrightarrow l(I_1) \geq b - a$$ ...

My thoughts ... we should be able to use $$\displaystyle (a, b) \subset [a, b]$$ and the fact that if $$\displaystyle A \subset B$$ then $$\displaystyle \mid A \mid \leq \mid B \mid$$ ... ... ... ... but we may have to prove rigorously that $$\displaystyle \mid (a, b) \mid = b - a$$ but how do we express this proof ...

Help will be much appreciated ... ...

Peter

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Readers of the above post may be assisted by access to Axler's definition of the length of an open interval and his definition of outer measure ... so I am providing access to the relevant text ... as follows:

Hope that helps ...

Peter

Last edited:

#### GJA

##### Well-known member
MHB Math Scholar
Hi Peter ,

Since $I_{1}$ is an open interval containing $[a,b]$, its right endpoint, say $c$, must be larger than $b$ (with infinity being allowed). Similarly, its left endpoint, say $d$, must be less than $a$ (with negative infinity being allowed). Can you proceed from here?

#### Peter

##### Well-known member
MHB Site Helper
Hi Peter ,

Since $I_{1}$ is an open interval containing $[a,b]$, its right endpoint, say $c$, must be larger than $b$ (with infinity being allowed). Similarly, its left endpoint, say $d$, must be less than $a$ (with negative infinity being allowed). Can you proceed from here?

Thanks for the help GJA ... appreciate it ... I believe I can now proceed ...

Now ... we wish to show that $$\displaystyle [a, b] \subset I_1 \Longrightarrow l(I_1) \geq b - a$$ ...

... so ... proceed as follows ...

Let $$\displaystyle I_1 = (c, d)$$ where $$\displaystyle c \leq a \lt b \leq d$$ ...

Then $$\displaystyle l(I_1) = d - c \geq b - a$$ ...

Is that correct?

Peter

GJA

#### GJA

##### Well-known member
MHB Math Scholar
Looks good. Nicely done!