# Theory of measurement: is a maximum set of rectangles known for which the Lebesgue differentiation theorem is met?

The Lebesgue differentiation theorem states that if $$x$$ it's a point in $$mathbb {R} ^ n$$ Y $$f: mathbb {R} ^ n rightarrow mathbb {R}$$ It is an integrable function of Lebesgue, then the limit of $$frac { int_B f d lambda} { lambda (B)}$$ on all the balls $$B$$ focused on $$x$$ as the diameter of $$B$$ going to $$0$$ it's the same almost everywhere for $$f (x)$$. But if you replace balls with other types of game with diameter goes to $$0$$This does not have to be true. For example, it does not have to be true if you replace balls with rectangles, even rectangles with sides are parallel to the coordinate axes.

But if we restrict things to $$L ^ p$$ functions where $$1 , then the Lebesgue differentiation theorem is valid for rectangles with sides parallel to the coordinate axes, but it is not valid for arbitrary rectangles. So my question is, it's a maximum set of known addresses for which the Lebesgue differentiation theorem is held $$L ^ p$$ Functions for the collection of all rectangles oriented in those directions?

Now this newspaper says that there is no known maximum set of addresses for tubes in $$mathbb {R} ^ n$$ for which it supports the Lebesgue differentiation theorem (although the document makes a conjecture about it). And a rectangle is nothing more than a tube in $$mathbb {R} ^ n$$. But it is possible that the answer is known for this special case.