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## Main Question or Discussion Point

Can someone show me an example to clarify this statement from Royden's Real Analysis:

The Lebesgue measure restricted to the sigma-algebra of Borel sets is not complete.

Now, from the definition of a complete measure space, if B is an element of space M, and measure(B) = 0, and A subset of B, then A is an element of M.

But my understanding of the Borel sets is that it is the smallest algebra containing all the open and closed sets. Hence A would be in the Borel set, hence A would be in M.

So I'm obviously missing something.

thanks

The Lebesgue measure restricted to the sigma-algebra of Borel sets is not complete.

Now, from the definition of a complete measure space, if B is an element of space M, and measure(B) = 0, and A subset of B, then A is an element of M.

But my understanding of the Borel sets is that it is the smallest algebra containing all the open and closed sets. Hence A would be in the Borel set, hence A would be in M.

So I'm obviously missing something.

thanks