Articles of singular measures

A sequence of singular measures converging weakly* to a continuous measure

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?

Singular measures on Real line

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.

Do there exist two singular measures whose convolution is absolutely continuous?

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?

How is a singular continuous measure defined?

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 […]

Singular continuous functions

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 […]