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On the relation between null sets and meagre sets, you can also look at this article. Two theorems mentioned in this note (both classical and not due to the author):

1. (As already mentioned above) There exist a meagre $F_\sigma$ subset $A$ and a null $G_\delta$ subset $B$ of $\mathbb R$ that satisfy $A\cap B=\emptyset$ and $A\cup B=\mathbb R$.

2. (The Erdős-Sierpiński Duality Theorem) Assume that the Continuum Hypothesis holds. Then there exists an involution (bijection of order two) $f:\mathbb R\to\mathbb R$ such that $f[A]$ is meagre if and only if $A$ is null, and $f[A]$ is null if and only if $A$ is meagre for every subset $A$ of $\mathbb R$.

While (1) says that the ideals of null, respectively meager sets are "orthogonal", (2) says that assuming CH they behave identically. But it is well known that this duality between measure and category fails dramatically once we take a more abstract point of view: Shelah proved that you need large cardinals to construct a model of set theory (ZF, no axiom of choice) where every set of reals is Lebesgue measurable, but no large cardinals are necessary to construct a model where every set of reals has the Baire property (the corresponding notion to measurability for category).

On the relation between null sets and meagre sets, you can also look at this article. Two theorems mentioned in this note (both classical and not due to the author):

1. (As already mentioned above) There exist a meagre $F_\sigma$ subset $A$ and a null $G_\delta$ subset $B$ of $\mathbb R$ that satisfy $A\cap B=\emptyset$ and $A\cup B=\mathbb R$.

2. (The Erdős-Sierpiński Duality Theorem) Assume that the Continuum Hypothesis holds. Then there exists an involution (bijection of order two) $f:\mathbb R\to\mathbb R$ such that $f[A]$ is meagre if and only if $A$ is null, and $f[A]$ is null if and only if $A$ is meagre for every subset $A$ of $\mathbb R$.

While (1) says that the ideals of null, respectively meager sets are "orthogonal", (2) says that assuming CH they behave identically. But it is well known that this duality between measure and category fails dramatically once we take a more abstract point of view: Shelah proved that you need large cardinals to construct a model of set theory (ZF, no axiom of choice) where every set of reals is Lebesgue measurable, but no large cardinals are necessary to construct a model where every set of reals has the Baire property (the corresponding notion to measurability for category).

On the relation between null sets and meagre sets, you can also look at this article. Two theorems mentioned in this note (both classical and not due to the author):

1. (As already mentioned above) There exist a meagre $F_\sigma$ subset $A$ and a null $G_\delta$ subset $B$ of $\mathbb R$ that satisfy $A\cap B=\emptyset$ and $A\cup B=\mathbb R$.

2. (The Erd\H os-Sierpi'nski Duality Theorem) Assume that the Continuum Hypothesis holds. Then there exists an involution (bijection of order two) $f:\mathbb R\to\mathbb R$ such that $f[A]$ is meagre if and only if $A$ is null, and $f[A]$ is null if and only if $A$ is meagre for every subset $A$ of $\mathbb R$.

While 1. says that the ideals of null, respectively meager sets are "orthogonal",  2. says that assuming CH they behave identically. But it is well known that this duality between measure and category fails dramatically once we take a more abstract point of view: Shelah proved that you need large cardinals to construct a model of set theory (ZF, no axiom of choice) where every set of reals is Lebesgue measurable, but no large cardinals are necessary to construct a model where every set of reals has the Baire property (the corresponding notion to measurability for category).

On the relation between null sets and meagre sets, you can also look at this article. Two theorems mentioned in this note (both classical and not due to the author):

1. (As already mentioned above) There exist a meagre $F_\sigma$ subset $A$ and a null $G_\delta$ subset $B$ of $\mathbb R$ that satisfy $A\cap B=\emptyset$ and $A\cup B=\mathbb R$.

2. (The Erdős-Sierpiński Duality Theorem) Assume that the Continuum Hypothesis holds. Then there exists an involution (bijection of order two) $f:\mathbb R\to\mathbb R$ such that $f[A]$ is meagre if and only if $A$ is null, and $f[A]$ is null if and only if $A$ is meagre for every subset $A$ of $\mathbb R$.

While (1) says that the ideals of null, respectively meager sets are "orthogonal", (2) says that assuming CH they behave identically. But it is well known that this duality between measure and category fails dramatically once we take a more abstract point of view: Shelah proved that you need large cardinals to construct a model of set theory (ZF, no axiom of choice) where every set of reals is Lebesgue measurable, but no large cardinals are necessary to construct a model where every set of reals has the Baire property (the corresponding notion to measurability for category).

On the relation between null sets and meagre sets, you can also look at this article. Two theorems mentioned in this note (both classical and not due to the author):

1. (As already mentioned above) There exist a meagre $F_\sigma$ subset $A$ and a null $G_\delta$ subset $B$ of $\mathbb R$ that satisfy $A\cap B=\emptyset$ and $A\cup B=\mathbb R$.

2. (The Erd\H os-Sierpi'nski Duality Theorem) Assume that the Continuum Hypothesis holds. Then there exists an involution (bijection of order two) $f:R\to R$ such that $f[A]$ is meagre if and only if $A$ is null, and $f[A]$ is null if and only if $A$ is meagre for every subset $A$ of $\mathbb R$.

While 1. says that the ideals of null, respectively meager sets are "orthogonal", 2. says that assuming CH they behave identically. But it is well known that this duality between measure and category fails dramatically once we take a more abstract point of view: Shelah proved that you need large cardinals to construct a model of set theory (ZF, no axiom of choice) where every set of reals is Lebesgue measurable, but no large cardinals are necessary to construct a model where every set of reals has the Baire property (the corresponding notion to measurability for category).

On the relation between null sets and meagre sets, you can also look at this article. Two theorems mentioned in this note (both classical and not due to the author):

1. (As already mentioned above) There exist a meagre $F_\sigma$ subset $A$ and a null $G_\delta$ subset $B$ of $\mathbb R$ that satisfy $A\cap B=\emptyset$ and $A\cup B=\mathbb R$.

2. (The Erd\H os-Sierpi'nski Duality Theorem) Assume that the Continuum Hypothesis holds. Then there exists an involution (bijection of order two) $f:\mathbb R\to\mathbb R$ such that $f[A]$ is meagre if and only if $A$ is null, and $f[A]$ is null if and only if $A$ is meagre for every subset $A$ of $\mathbb R$.

While 1. says that the ideals of null, respectively meager sets are "orthogonal", 2. says that assuming CH they behave identically. But it is well known that this duality between measure and category fails dramatically once we take a more abstract point of view: Shelah proved that you need large cardinals to construct a model of set theory (ZF, no axiom of choice) where every set of reals is Lebesgue measurable, but no large cardinals are necessary to construct a model where every set of reals has the Baire property (the corresponding notion to measurability for category).