Can anyone provide a sequence of singular (w.r.t. Lebesgue measure) measures $\in\mathcal{M}([0,1])=C[0,1]^*$ converging $weakly^*$ to an absolutely continuous (w.r.t. Lebesgue measure) measure?

Can some one please give me an example of continuous Singular measure on Reals which is not absolutely continuous to cantor-type sets. Thank you.

Let $\mu, \nu$ be finite complex measures with compact supports on the real line, and assume that they are singular with respect to the Lebesgue measure. Can their convolution $\mu\ast\nu$ have a nonzero absolutely continuous component?

On a measurable space, how is a measure being singular continuous relative to another defined? I searched on the internet and in some books to no avail and it mostly appears in a special case – the Lebesgue measure space $\mathbb{R}$. Do you know if singular continuous measures can be generalized to a more general […]

A function $f:[0,1]\rightarrow\mathbb{R}$ is called singular continuous, if it is nonconstant, nondecreasing, continuous and $f^\prime(t)=0$ whereever the derivative exists. Let $f$ be a singular continuous function and $T$ the set where $f$ is not differentiable. Question: Is $T$ nowhere dense? Examples: A classical example of such a function is the so-called devil’s staircase, obtained as […]

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