如何构造具有正测度的连通无处稠密集或平面曲线？

Hey there! No worries at all about your English—let’s break down how to construct those sets you’re curious about. Here are some practical, intuitive approaches:

The standard Cantor set is nowhere dense but has measure zero, so we need to tweak the construction to keep connectivity while boosting the measure.

• Modified "Fat Cantor" Construction
  Instead of cutting out intervals that split the remaining set into disconnected pieces, we remove open intervals only from one "side" of the remaining set to preserve connectivity. For example:

  1. Start with the interval [0, 1].
  2. First, remove (1/2, 3/4) (length removed: 1/4). Then from [0, 1/2], remove (1/4, 3/8) (length removed: 1/8), then from [0, 1/4] remove (1/8, 3/16), and so on.
  • The total length removed is 1/4 + 1/8 + 1/16 + ... = 1/2, so the remaining set has measure 1 - 1/2 = 1/2 > 0.
  • This set is connected: it contains 0 and 1, and any point in the set can be connected to 1 via the rightmost remaining segments (like [3/4,1], [7/8,1], etc.). It's also nowhere dense—any open subinterval of [0,1] will contain part of one of the removed open intervals, so the set has no interior points.

• Alternative: Using Continuous Functions
  You can also construct such a set as the domain of a continuous, non-decreasing function that's constant on a sequence of intervals (like a generalized Cantor function), but where the total length of the constant intervals is less than the original interval's length. The remaining domain (where the function is strictly increasing) will be a connected, nowhere dense set with positive measure.

The classic example here is the Osgood curve, a continuous plane curve whose image has positive 2-dimensional measure. Here's how to think about its construction:

• Iterative Connected Set Construction
  We build a sequence of connected, closed sets whose intersection is the curve's image, ensuring each step preserves connectivity and keeps the total area positive:

  1. Start with the unit square S₀ = [0,1]×[0,1].
  2. Split S₀ into 9 smaller squares of side length 1/3. Remove one edge square (not the middle one) and connect the remaining 8 squares with adjacent links to keep S₁ connected. The area of S₁ is 8/9.
  3. Adjust the removal ratio to keep final measure positive: instead of removing 1/9 of each square's area, remove only 1/100. The total area removed becomes a convergent series, leaving a positive measure for the intersection set.
  4. The intersection of all these nested Sₙ is the Osgood curve's image: it's connected (nested connected closed sets in a complete space have a connected intersection), it's the image of a continuous map from [0,1] (via extending a map from the Cantor set), and it has positive 2-dimensional measure.

• Key Takeaways for Both Cases

  • Connectivity: Always ensure each construction step keeps the set connected—nested connected closed sets in a complete space have a connected intersection.
  • Positive Measure: Control the total "size" of removed regions so their sum is less than the original set's measure.
  • Nowhere Dense: Make sure no open ball/interval is entirely contained in the final set—this happens naturally if you remove open regions in every possible subinterval/ball during construction.

内容的提问来源于stack exchange，提问作者Mark Rakhel