Actually I am learning about measure theory. But I have confusion between topological space and measurable . Is there any relationship among them or not?
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$\begingroup$The two are unrelated mathematical structures. However, every topological space can be given a particular useful $\sigma$-algebra, called the Borel $\sigma$-algebra. This is the smallest $\sigma$-algebra containing all the open sets of the space. One useful property is that continuous real-valued functions are measurable with respect to this $\sigma$-algebra. More generally, if $X,Y$ are topological spaces, and you equip them both with their Borel $\sigma$-algebras, then continuous functions from $X$ to $Y$ are measurable.
The most commonly used $\sigma$-algebras are the Lebesgue measurable subsets of $\mathbb{R}^n$, for each $n$. These are not the Borel $\sigma$-algebras of these spaces. However, every Lebesgue measurable set is the union of a Borel set and a subset of a Borel set of measure zero.
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