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$R$ is the union of a meager set and a set of measure zero, so smallness of category and smallness of measure are independent notions

Statement refuted

Refuted claim: meagreness and measure zero are comparable notions of smallness, so that a set small in one sense is small, or at least not co-small, in the other.

The witness is a decomposition $R = (R ∖ G) \cup G$ in which $R ∖ G$ is meager (Nowhere dense, meager (first category), residual, and second category subsets of $R$ ) and $G$ has measure zero (Measure zero (a countable cover by intervals of total length below every $ε$ ) and content zero (a finite such cover)). So $R$ , which by Baire category in $R$ , by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so $R$ is not a countable union of nowhere dense sets is not meager, splits into two pieces each of which is negligible, one in the sense of category and one in the sense of measure. In particular $G$ is residual and null at the same time, and its complement is meager and, being the complement of a null set, in no sense small in measure.

The set is $G : = ⋂_{n \in N} U_{n}, U_{n} : = ⋃_{k \in N} (e (k) - 2^{- k - n - 2}, e (k) + 2^{- k - n - 2}),$ where $e : N \to Q_{R}$ is a bijection onto the rationals (The rationals embed densely in the reals).

Facts & Assumptions

Given: A bijection $e : N \to Q_{R}$ onto the rationals inside $R$ , the sets $U_{n}$ and $G$ displayed above.

[A1]

The refuted claim: a meager set and a set of measure zero cannot together exhaust $R$ , meagreness and nullity being comparable notions of smallness.

[L1]

$Q \approx N$ and $q \mapsto \hat{q}$ is injective onto $Q_{R}$ , so a bijection $e$ exists; $Q_{R}$ is dense in $R$ ( $Q$ is countably infinite, Equinumerous sets, $A \approx B$ and $A ⪯ B$ , Finite, countably infinite, countable, uncountable, A nonempty set is at most countable iff it is a surjective image of $N$ , The rationals embed densely in the reals, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable, Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ).

[L2]

$(c, d)$ is an open interval of length $d - c$ and is an open set; an arbitrary union of open sets is open; the complement of an open set is closed (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Open subset of $R$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen, Arbitrary unions and finite intersections of open subsets of $R$ are open, and dually for closed sets).

[L3]

A set is dense exactly when every $N_{ε} (x)$ meets it; a closed set is nowhere dense exactly when its interior is empty; a meager set is a union of a sequence of nowhere dense sets (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ , The closure equals the set together with its limit points, equals the set of points every neighbourhood of which meets it, and is the smallest closed superset; a set is closed iff it contains its limit points, The $ε$ -neighbourhood and the punctured $ε$ -neighbourhood of a point of $R$ , Nowhere dense, meager (first category), residual, and second category subsets of $R$ , Interior, closure, boundary and exterior of a subset of $R$ ).

[L4]

$\sum_{k = 0}^{\infty} 2^{- k} = 2$ , powers satisfy $2^{- k - n - 1} = 2^{- n - 1} 2^{- k}$ , finite sums scale, and $2^{- n} \to 0$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, Series, partial sums, convergence and the sum, divergence, and the tail series, Integer powers $a^{m}$ , Laws of integer exponents, Finite sums and finite products, by recursion, Laws of finite sums and finite products, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum, For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , Limits and Cauchy sequences of reals).

[L5]

Nullity: $A$ is null when for every real $ε > 0$ there is a sequence of closed intervals covering $A$ with all partial total lengths at most $ε$ ; every at most countable set is null (Measure zero (a countable cover by intervals of total length below every $ε$ ) and content zero (a finite such cover), Every at most countable subset of $R$ has measure zero).

[L6]

$R$ is not a union of a sequence of nowhere dense sets (Baire category in $R$ , by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so $R$ is not a countable union of nowhere dense sets).

[L7]

$G$ is an intersection of a sequence of open sets, so it is $G_{δ}$ ( $F_{σ}$ and $G_{δ}$ subsets of $R$ ).

[L8]

Ordered-field arithmetic: $0 < 1$ , so $2 > 0$ and $2^{- k - n - 2} > 0$ ; adding a constant and multiplying by a positive preserve an inequality (The multiplicative identity is positive, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.

Counterexample

technique · direct

1.1

Fix $e$ by [L1]. Each $U_{n}$ is an open set, being a union of open intervals by [L2], and each contains $Q_{R}$ , since $e (k)$ lies in the $k$ -th interval by [L8]. Hence each $U_{n}$ is dense by [L1] and [L3], a superset of a dense set being dense.

givenL1L2L3L8choose

2.1

$G$ is null. Let the real $ε > 0$ be given and use [L4] to fix $n$ with $2^{- n} \leq ε$ . The closed intervals $I_{k} : = [e (k) - 2^{- k - n - 2}, e (k) + 2^{- k - n - 2}]$ cover $U_{n}$ , hence cover $G \subseteq U_{n}$ , and each has length $2^{- k - n - 1} = 2^{- n - 1} 2^{- k}$ by [L2], [L4] and [L8]; so every partial total length is $\sum_{k < i} 2^{- n - 1} 2^{- k} = 2^{- n - 1} \sum_{k < i} 2^{- k} \leq 2^{- n - 1} \cdot 2 = 2^{- n} \leq ε$ by [L4]. By [L5] the set $G$ has measure zero.

step 1.1L4L5L8

2.2

$R ∖ G$ is meager. By De Morgan $R ∖ G = ⋃_{n} (R ∖ U_{n})$ , and each $R ∖ U_{n}$ is closed by [L2]. Its interior is empty: if $N_{δ} (x) \subseteq R ∖ U_{n}$ for some real $δ > 0$ , then $N_{δ} (x)$ would miss $Q_{R} \subseteq U_{n}$ , contradicting the density of $Q_{R}$ given by [L1] and [L3]. So each $R ∖ U_{n}$ is nowhere dense by [L3], and $R ∖ G$ is meager.

step 1.1L1L2L3

3.1

So $R = (R ∖ G) \cup G$ with the first piece meager and the second null, which is the failure of [A1]. Moreover $G$ is residual, its complement being meager, and $G \neq \emptyset$ : were $G$ empty, $R = R ∖ G$ would be meager, contradicting [L6]. Thus $G$ is a residual, $G_{δ}$ set of measure zero by [L7], and $R ∖ G$ is a meager set whose complement is null.

step 2.1step 2.2A1L3L6L7∎

Remarks

Both pieces are as small as their notion allows, and they are complementary. $G$ is null and residual; $R ∖ G$ is meager and its complement is null. So no implication holds between "meager" and "null" in either direction, and neither can be strengthened to a statement about the complement. This is the standard duality between measure and category, and $G$ is the standard witness for it.
$G$ contains all the irrationals that are well approximable by rationals. Membership in $U_{n}$ says that some rational $e (k)$ lies within $2^{- k - n - 2}$ of the point, so $G$ is a set of points approximable by rationals at every accuracy of that shape. Nothing on this page needs that reading; it is recorded because it is what makes the example natural rather than contrived.
Baire is used only once, and only for nonemptiness. Steps 2.1 and 2.2 are independent of Baire category in $R$ , by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so $R$ is not a countable union of nowhere dense sets; it enters in step 3.1 to rule out $G = \emptyset$ , which would make the decomposition vacuous. That is also the precise sense in which the example needs the completeness of $R$ .
The individual $U_{n}$ are open, dense and of small total cover length, which is $Q$ is covered by open intervals of total length $ε$ , for every $ε > 0$ with $ε = 2^{- n}$ ; the example is that construction iterated and intersected.

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Sources

Meagre set (Wikipedia) (standard reference, not scraped)
Null set (Wikipedia) (standard reference, not scraped)
J. C. Oxtoby, Measure and Category, 2nd ed., Ch. 1-2 (John C. Oxtoby) (standard reference, not scraped)
Meager set (Encyclopedia of Mathematics) (standard reference, not scraped)

RR is the union of a meager set and a set of measure zero, so smallness of category and smallness of measure are independent notions