A non-measurable set is a type of set in mathematics that cannot be assigned a meaningful size or measure using standard methods. This concept arises in the field of set theory and measure theory, where certain sets defy the usual rules of measurement, such as the Lebesgue measure. An example of a non-measurable set is the Vitali set, which is constructed using the properties of real numbers. Non-measurable sets challenge our intuitive understanding of size and volume. They often emerge in discussions about infinity and the continuum hypothesis, highlighting the complexities of real analysis. These sets illustrate that not all collections of points can be quantified, leading to deeper insights in mathematical theory.