Clarification on the countable additivity of Lebesgue measure

As emphasized in remark 1.2.4, p. 19, of Terence Tao’s book “An introduction to measure theory”, finite additivity doesn’t hold for Lebesgue outer measure m^{*}(\cdot) in general, and therefore it doesn’t hold for Lebesgue measure m(\cdot) either. So, the Lebesgue outer measure m^{*}(E\cup F) of the union of two disjoint sets E, F in the Euclidean metric space (\mathbb{R}^d, |\cdot|) does not necessarily satisfy m^{*}(E\cup F)=m^{*}(E)+m^{*}(F).

If E,F are both Lebesgue measurable, then it holds that m^{*}(E\cup F)=m^{*}(E)+m^{*}(F). Moreover, m^{*}(E)=m(E)m^{*}(F)=m(F), which means that the union E\cup F is also Lebesgue measurable and ultimately finite additivity follows as m(E\cup F)=m(E)+m(F).

The main point is that if E\cup F is Lebesgue measurable and E\cap F=\emptyset, then finite additivity doesn’t follow. Instead, the disjoint set assumption E\cap F=\emptyset needs to be replaced by the positive distance assumption \mbox{dist}(E, F)=\inf\{|x-y|: x\in E, y\in F\}>0 to ensure finite additivity for the outer measure, as explained and proved in lemma 1.2.5, p. 19, of Tao’s book.

Considering this limitation in the applicability of finite additivity, a reader may feel alarmed when reading lemma 1.9.c, p. 21, of David Williams’ book “Probability with martingales”, which states that if m(S)<\infty in a measure space (S, \Sigma, m), then m(F\cup E) = m(F)+m(E)-m(F\cap E) for F, E\in\Sigma. If F\cap E=\emptyset, then m(F\cup E) = m(F)+m(E), so finite additivity holds.

Recall that a measure m:\Sigma\rightarrow[0,\infty] on (S,\Sigma) has the whole σ-algebra \Sigma on S as its domain. This implies that for any F, E\in\Sigma the measures m(F),m(E) exist by assumption, therefore there is no conflict between the aforementioned statements on finite additivity of measure found in Tao’s and Williams’ book.

It becomes clear that operating on a measure space (\mathbb{R}^d,\Sigma, m) seems to avoid the trouble of having to prove the existence of Lebesgue measure for every set of interest, simply because by definition m(F) exists for every F\in\Sigma. However, there is no free lunch. A follow-up question arises, as one then needs to show that F belongs to the defined σ-algebra \Sigma, which is not always trivial.

For instance, the experiment of tossing a coin infinitely often is presented in p. 24 of Williams’ book by introducing an associated sample space \Omega and subsequently a probability triple (\Omega,\mathcal{F},P). An event F of possible interest is that the ratio of heads in n tosses tends to 1/2 as n\rightarrow\infty. Even for such a seemingly simple experiment, proving that this event F, seen as a set, belongs to the σ-algebra \mathcal{F} on \Omega is already not so trivial to prove.

On a final note, lemma 1.2.15, p. 30, in Tao’s book, proves countable additivity for disjoint Lebesgue measurable sets, which subsumes finite additivity. The proof is easy to follow; the claim is proved first for compact, then for bounded and then for unbounded sets. To conclude the present post, a clarification is made in the proof of the bounded case. In particular, it will be explained why for a bounded Lebesgue measurable set E_n there exists a compact set K_n such that m(E_n) \le m(K_n)+\epsilon/2^n.

Since E_n is Lebesgue measurable, it follows from exercise 1.2.7.iv in Tao’s book that for any \epsilon > 0 there exists a closed set K_n such that K_n\subseteq E_n and m^{*}(E_n\setminus K_n) \le \epsilon/2^n. By applying the countable subadditivity of outer measure (see exercise 1.2.3.iii in Tao’s book), E_n=K_n\cup (E_n\setminus K_n) leads to m^{*}(E_n)\le m^{*}(K_n)+m^{*}(E_n\setminus K_n) \le m^{*}(K_n)+\epsilon/2^n. Furthermore, K_n is Lebesgue measurable since it is closed (see lemma 1.2.13.ii in Tao’s book). Thus, Lebesgue outer measure can be replaced by Lebesgue measure to give m(E_n) \le m(K_n)+\epsilon/2^n. Finally, K_n is bounded, as a subset of the bounded set E_n, and closed, so K_n is compact, which completes the proof.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s