False beliefs about Lebesgue measure on $\mathbb{R}$

Question

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is countable, the border of a set has measure zero, etc. Can you help me sharing your experience or with some reference list?

Answer 1

More Cantor madness:

True belief:

There is a measurable set $A$ in $[0,1]$ such that for any interval $U$ in $[0,1]$, both $A\cap U$ and $A^c\cap U $ have positive measure.

False belief:

The continuous image of a set of measure 0 has measure 0.

Answer 2

False belief: the continuous image of a measurable set is measurable.

A counterexample is provided by the Devil's staircase. Since the image of the Cantor set has full measure, it will have subsets, still measurable, which have non-measurable image. The same function also serves as a counterexample to the following:

False belief: if a continuous function has derivative zero almost everywhere, then it is constant.

Answer 3

False belief: a subset of an interval that is both open and dense has the measure of the interval.

A counterexample is obtained by enumerating the rationals on $[0,1]$ and putting an open interval of length $(1/3)^k$ around the $k$th one. The union of these intervals is clearly dense because it contains a dense set (the rationals) as a subset, and it is clearly open because it is a union of open intervals. But meanwhile, its Lebesgue measure is $\leq \sum_1^\infty (1/3)^k = 1/2$.

Answer 4

False Belief: A nowhere dense subset of $\mathbb{R}$ has measure $0$. (Let me recall that a subset $A$ of $\mathbb{R}$ is said to be nowhere dense if the interior of its closure is empty.)

I leave the explanation as to why this is indeed a false belief as an exercise!

Answer 5

Consider the following (true) statement:

If $(I_n)$ is a sequence of subintervals of the unit interval and the sum of their lengths is strictly less than $1$, then the $I_n$ do not cover the unit interval.

False belief: This can be proven just by translating $I_1$ to begin at $0$, translating $I_2$ to start end the end of $I_1$ etc. If this worked, then the same would be true for the unit interval in $\mathbb Q$ where the statement is false.

I obvious can't claim this to be original; I got it from MO.