By Amanda.M

2015-02-28 14:52:45 8 Comments

I am working on this problem$^{(1)}$ on Lebesgue measurability of composition of Lebesgue measurable function and a continuous function:

Show that $g \circ f$ is Lebesgue measurable, if $f: X \to \mathbb R$ is Lebesgue measurable and if $g: \mathbb R \to \mathbb R$ is continuous.

Prior to this posting, I did lots of research online. First, I found this 6-year-old solution to a problem very similar to mine here at MathHelpForum, but on closer inspection I think not only the author left lots of gap, but also I am not sure if this is a correct solution, and on top of it I do not really understand it. And then internally in MSE, I found this 2012's posting and also this 2013's posting, which are similar but not exactly the same.

In my naive logic, I am thinking of first proving that $g$ is measurable because of its continuity, and then since composition of 2 measurable functions is measurable, therefore $g \circ f$ is measurable. But my logic is unreliable, please help me with the right direction and also steps to solve this question.

Thank you very much for your time and help.

I forget to include definition to Lebesque measurability and Lebesgue measurable function until @PhoemueX brought it up. The problem with this text is that it does not have one nice, stand-alone paragraph definition. According to its index, it is written here and there on pages 21, 27 and 39. Here is what I managed to piece them together from those pages:

Let $X = \mathbb R$ and let $\mathcal C$ be the collection of intervals of the form $(a, b]$... Let $\mathcal l(I) = b - a$ if $I = (a, b]$... Define $\mu^*$ as an outer measure... however, that if we restrict $\mu^*$ to a $\sigma$-algebra $\mathcal L$ which is strictly smaller than the collection of all subsets of $\mathbb R$, then $\mu^*$ will be a measure on $\mathcal L$. That measure is what is known as Lebesgue measure. The $\sigma$-algebra $\mathcal L$ is called the Lebesgue $\sigma$-algebra.... A set is Lebesgue measurable if it is in the Lebesgue $\sigma$-algebra.

If $X$ is a metric space, $\mathcal B$ is the Borel $\sigma$-algebra, and $f: X \to \mathbb R$ is measurable with respect to $\mathcal B$, we say $f$ is Borel measurable. If $f : \mathbb R \to \mathbb R$ is measurable with respect to the Lebesgue $\sigma$-algebra, we say $f$ is Lebesgue measurable function.

(1) Richard F. Bass' Real Analysis, 2nd. edition, chapter 5: Measurable Functions, Exercise 5.6, page 44.


@Clement C. 2015-02-28 16:02:37

Edit: following the comment of kahen below, I modified my answer: Lebesgue-measurability of a function $h$ is measurability of $h\colon (X,\mathcal{L}_X)\to (Y,\mathcal{B}_{Y})$, not $h\colon (X,\mathcal{L}_X)\to (Y,\mathcal{L}_{Y})$)

You have that $f\colon (X,\mathcal{L}_X)\to (\mathbb{R},\mathcal{B}_\mathbb{R})$ is Lebesgue-measurable (for the $\sigma$-algebras $\mathcal{L}_X$, $\mathcal{B}_\mathbb{R}$). As $g\colon \mathbb{R}\to \mathbb{R}$ is continuous, it is Borel-measurable ($g\colon (\mathbb{R},\mathcal{B}_\mathbb{R})\to (\mathbb{R},\mathcal{B}_\mathbb{R})$ is measurable for the $\sigma$-algebras $\mathcal{B}_\mathbb{R}$, $\mathcal{B}_\mathbb{R}$). You want to show that $g\circ f$ is Lebesgue-measurable. i.e. $g\circ f \colon (X,\mathcal{L}_X)\to (\mathbb{R},\mathcal{B}_\mathbb{R})$ is measurable.

Take any $B\in\mathcal{B}_\mathbb{R}$: you need to show that $(g\circ f)^{-1}(B)\in \mathcal{L}_X$.

By measurability of $g$, you have that for since $B\in\mathcal{B}_\mathbb{R}$, $B^\prime = g^{-1}(B)\in \mathcal{B}_\mathbb{R}$. By measurability of $f$, this implies that $f^{-1}(B^\prime)\in \mathcal{L}_X$, i.e. $(g\circ f)^{-1}(B)\in \mathcal{L}_X$. This shows that $g\circ f$ is measurable for the $\sigma$-algebras $\mathcal{L}_X$, $\mathcal{B}_\mathbb{R}$ (i.e., $g\circ f\colon (X, \mathcal{L}_X)\to (\mathbb{R}, \mathcal{B}_\mathbb{R})$ is measurable), as wanted.

@Clement C. 2015-02-28 16:13:30

(If you define $f$ being Lebesgue-measurable as being measurable from $(X,\mathcal{L}_X)$ to $(\mathbb{R},\mathcal{B}_\mathbb{R})$, then it is straightforward, though. How was it defined in your book?)

@kahen 2015-02-28 16:17:52

You definitely do not give the codomain the Lebesgue $\sigma$-algebra. The problem is that there are continuous maps which are not $\mathcal L_{\mathbb R}$-$\mathcal L_{\mathbb R}$ measurable.

@Clement C. 2015-02-28 16:19:15

OK. So if Lebesgue-measurability of $h\colon X\to Y$ is measurability of $h\colon (X,\mathcal{L}_X)\to (Y,\mathcal{B}_Y)$, then the above works...

@Clement C. 2015-02-28 16:26:26

(regarding your last point, yes -- that was what confused me regarding the validity of the question, for the definition of Lebesgue-measurability I was assuming)

@Amanda.M 2015-02-28 16:39:30

@ClementC. : I have post-scripted whatever the text has on the definition of Lebesgue measurability, see above. Let me know and thanks as always.

@Clement C. 2015-02-28 16:40:26

From what I gather (I also had a look at the book itself), this is consistent with the definition in the "Edit" part.

@Amanda.M 2015-02-28 16:50:40

Thanks, I am still digesting your solution bit by bit...

@Amanda.M 2015-02-28 18:48:05

@ClementC. I like your solution better! Your definition of measurable function, i.e., $(g \circ f)^{-1}(B) \in \mathcal L_X$ mimics that of Wikipedia's: . But my text uses a "different" definition: "A function $f : X \to \mathbb R$ is measurable or $\mathcal A$-measurable if $\{x \mid f(x) > a\} \in \mathcal A, \forall a \in \mathbb R$." Wikipedia's & yours is more intuitive. Are they the same or textbook-specific? (I am talking about measurable fuction, not Lebesgue measurable function.) Thanks again and again.

@Clement C. 2015-02-28 19:12:12

@A.Magnus: they are the same: yours essentially looks at sets $B_a=(a,+\infty)$, and considers $f^{-1}(B_a)$. But it is a classic result that the $\sigma$-algebra generated by $(B_a)_{a\in\mathbb{R}}$ is exactly $\mathcal{B}_\mathbb{R}$

@Amanda.M 2015-02-28 19:34:02

Thanks, quite relieved hearing they are the same! The same question then goes on asking (a) Is this also true if $g$ is Borel measurable instead of continuous? (b) Is this true if $g$ is Lebesgue measurable instead of continuous? I think the answer for (a) is yes and for (b) is no? Can you shed some more light here? Thanks again and again.

@Ian 2015-02-28 20:23:47

@A.Magnus (a) Yes; the key is that $g^{-1}(A)$ is Borel when $A$ is Borel, so $f^{-1}(g^{-1}(A))$ is the preimage of a Borel set under $f$.

@Clement C. 2015-02-28 20:28:34

(b) Indeed, it is false. That's the key thing in the counterexample from here: if $g$ is only Lebesgue measurable, $B^\prime=g^{-1}(B)$ can end up being in $\mathcal{L}_\mathbb{R}\setminus \mathcal{B}_\mathbb{R}$, and then you don't have the guarantee that $f^{-1}(B^\prime)$ is in $\mathcal{L}_\mathbb{R}$.

@Ian 2015-02-28 20:29:10

@ClementC. I was in the process of writing down that example myself :)

@Amanda.M 2015-03-03 02:01:15

@ClementC. : Hope you get this message: Thanks for your helps and now this problem was completely done, couple of clarifications, if you please: (a) Just wanted to confirm that Lebesgue $\sigma$-algebra is always more inclusive than Borel $\sigma$-algebra, am I correct? (b) The hypothesis first says that $g : \mathbb R \to \mathbb R$ is continuous, and then you claim that $g: (\mathbb R, \mathcal B_R) \to (\mathbb R, \mathcal B_R)$ is measurable. How do you determine that $\mathbb R$ has $\mathcal B$ in the first place, even though hypothesis does not say it? Is it by default for $\mathbb R$?

@Clement C. 2015-03-03 03:41:43

Hi, Yes, one has $\mathcal{B}\subset \mathcal{L}$. As for continuity, it does imply Borel-measurability (this is a theorem), and Borel sets is a "antural" $\sigma$-algebra for $\mathbb{R}^n$)

@Amanda.M 2015-03-04 13:05:58

I think you meant "natural," not "anatural," correct? Thanks again and again, I green check-marked, up-voted your answer, and up-arrowed all you comments, hoping you get points bountiful thereafter.

@Clement C. 2015-03-04 13:31:55

(yes, natural... sorry)

@Ian 2015-02-28 16:08:23

Note that $ (g \circ f)^{-1}(A)=f^{-1}(g^{-1}(A))$. You need to show this is Lebesgue measurable if $ A $ is open. Now the key is that if $A$ is open, then so is $ g^{-1}(A) $. So $(g \circ f)^{-1}(A)=f^{-1}(B) $ where $ B $ is open. Conclude from there.

@kahen 2015-02-28 16:55:10

Borel instead of open, right?

@Ian 2015-02-28 17:28:43

@kahen It doesn't matter. The books I have used define measurability of real-valued functions just with open sets or even just with infinite rays. The $\sigma$-algebra properties and the fact that preimages commute with unions and intersections take care of the rest.

@Amanda.M 2015-02-28 19:34:47

@Ian : Thank you for your speedy response!

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