S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of $\mathrm{Homeo}([0,1])$ has length 2. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of $\mathrm{Homeo}([0,1])$ have length 2. We answer this question in the affirmative.

We call $f\in \mathrm{Homeo}({[0,1]}^d)$ singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of $f\in \mathrm{Homeo}([0,1])$ has length 2 iff f is singular iff f is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem.

We show that for $d\ge 2$ the graph of the generic element of $\mathrm{Homeo}({[0,1]}^d)$ has infinite d-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of $\mathrm{Homeo}({[0,1]}^d)$ with infinite d-dimensional Hausdorff measure is not Haar null. We show that for $d\ge 2$ the generic element of $\mathrm{Homeo}({[0,1]}^d)$ is singular but not strongly singular. We also show that for $d\ge 2$ almost every element of $\mathrm{Homeo}({[0,1]}^d)$ is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null).

Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them.

S. Banach指出， $\mathrm{荷美}（[0，\mathrm{1个}]）$ 长度为2。J. Mycielski询问测度理论对偶是否成立，即，除Haar以外的所有图是否都包含了（在Christensen的意义上）许多元素 $\mathrm{荷美}（[0，\mathrm{1个}]）$ 长度为2。我们肯定地回答这个问题。

我们称之为 $F\in \mathrm{荷美}（{[0，\mathrm{1个}]}^d）$如果对空集采取适当的一组完整量度，则为奇异值；如果几乎在任何地方都可与奇异导数矩阵微分，则为奇异奇异值。由于图$F\in \mathrm{荷美}（[0，\mathrm{1个}]）$如果f为奇异，且f为奇异，则长度为2 ，则以下结果是Banach观测的高维类似物以及我们对Mycielski问题的解决方案。

我们证明了 $d\ge \mathrm{2个}$ 的通用元素图 $\mathrm{荷美}（{[0，\mathrm{1个}]}^d）$具有无限的d维Hausdorff测度，与Banach的上述结果形成对比。度量理论对偶仍然是开放的，但是我们证明了$\mathrm{荷美}（{[0，\mathrm{1个}]}^d）$具有无限d维Hausdorff测度的结果不是Haar null。我们证明了$d\ge \mathrm{2个}$ 的通用元素 $\mathrm{荷美}（{[0，\mathrm{1个}]}^d）$是单数，但不是很单数。我们还证明了$d\ge \mathrm{2个}$ 几乎每个元素 $\mathrm{荷美}（{[0，\mathrm{1个}]}^d）$ 是奇异的，但强奇异元素的集合形成了所谓的Haar矛盾集（既不是Haar空值，也不是co-Haar空值）。

最后，为了澄清这种情况，我们研究了几个变量映射的奇异性的各种可能定义，并探讨了它们之间的联系。