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Q: Is there a Lebesgue integrable function $f$ such that the set of points on no neighborhood of which $f$ is bounded has positive Lebesgue measure? I tried to prove that there are none using the definition of the Lebesgue integral of nonnegative functions (by converting it into an improper Riemann integral of another variable, see Wikipedia for that) but don't know how to proceed.

A: Yes. Let $$g(t)=t^{-1/2}\chi_{(0,1)}(t),$$ say $(t_n)$ is a countable dense subset of $\Bbb R$, and define $$f(t)=\sum2^{-n}g(t-t_n).$$
Then $f$ is integrable but $f$ is unbounded in every nonempty open set.


The Central Market is always a hive of activity with a range of goods unique to the Solomon Islands. But have you ever wondered how they were made? Or where they came from?
In the furthermost part of the market from Iron Bottom Sound, the clothes sellers, jewellery and oil sellers set up shop. One such trader is Harriet Suiga.
She told SIBC the process behind creating the lavalava – a Solomon Islands trademark.
"I get the rolls (of material)," she said. "Each roll costs $175, and one roll will get about 8-10 lavalavas.
She sells the lavalavas for $35 each, with a Solomon Islands stencil, created by one of the stencil makers nearby, planted on them.
They are staples of the Solomon Islands; but what is the best kind of coconut to buy?
According to Sharon Teresa – and the ladies selling coconuts outside at the Central Market around her – the answer is: "all of them".
The group of coconut sellers buy them from farmers in the morning who come in from around the Guadalcanal area – both east and west – to the sell their produce. $150-$200 will get you "70 to 80 coconuts" depending on the size.
How many the women sell – usually for $5 each, with the small ones going for $3 – depends on the type of day it is.
"It's a mix of people," she said. "We get locals buying for their wantok. Tourists come, and I even get tourists coming in and buying the bride price necklaces," she said, in reference to the expansive decorations on her left, in the photo above, which take weeks to make.
The most popular were the smaller necklaces, but people loved looking at the big bride price necklaces, she said.
She normally spent six days a week at the stand, with busy periods generally coming on the weekends.


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Mon, 23 Apr 2012 | Intestinal Microbiota
a N, R, W, and Y are the International Union of Pure and Applied Chemistry codes for ambiguous bases.
and diet in the human populations studied. Two large studies, where an extensive array of oligonucleotide probes that targeted the major bacterial groups in the Gl-tract of northern European adults was used, showed that 62-75% of the fecal bacteria could be detected and identified. The remainder (z 30%) could either belong to members of the Archaea, Eukarya or most likely to yet unknown bacteria (90,91). These types of studies provide a valuable basis in order to eventually determine factors that change the microbiota such as lifestyle, diet or illness. Interestingly, FISH-FCM analysis of fecal microbiota of patients with UC revealed substantial temporal variations in the major bacterial groups studied (i.e., Bacteroides, C. coccoides-E. rectale, Atopobium, bifidobacteria and lactobacilli), which was further was supported by PCR-DGGE profiles (40).
NEW MOLECULAR DIVERSITY APPROACHES Real Time PCR
Real time quantitative PCR (qPCR) of the 16S rRNA gene is being developed the last few years for the detection and quantification of human intestinal microbiota, which has the advantages of being high throughput and measuring from 1 to up to 108 CFU (99). Both SYBR Green I and TaqMan chemistries have been used to target Bacteroides fragilis, Bifidobacterium species, E. coli, L. acidophilus and Ruminococcus productus, and the method was demonstrated to be easier and faster than dot-blot hybridization methodology (100). Real-time qPCR (50 nuclease PCR assay) has been used to study the microbiota that adhere to the colonic mucosa (101). The primer-probe combinations were applied to DNA for the detection of E. coli and Bacteroides vulgatus from pure cultures and colonic biopsy specimens. The assay was very sensitive detecting as little as 1 and 9 CFU of E. coli and B. vulgatus, respectively. Many of the qPCR assays being developed target the lactobacilli and Bifidobacterium species that may be incorporated in functional foods (102,103). Besides real time PCR of the 16S rRNA gene, the option to use the transaldolase gene of Bifidobacterium species has also been investigated and appeared to be superior to the former in quantifying bifidobacterial populations in infants (104). The qRT-PCR assays have been used for various applications such as comparison of healthy persons versus patients suffering from IBS (105), and in patients with active IBD (26). Recently, a TaqMan real-time PCR-based method for the quantification of 20 dominant bacterial species and groups of the microbiota was developed (106). This method involved a pair of conserved primers, as well as universal and specific quantification probes, for species, group or genus in question, in a single reaction, and allowed relative and absolute quantification of bacteria in human biopsy and fecal samples. Further developments in real-time qPCR will facilitate our insight into the dynamics of the microbiota.
Diagnostic DNA Microarrays
The development of DNA oligonucleotide microarrays offer a fast, high throughput option for detection and estimation of the diversity of microbes in a complex ecosystem (107). Alternative terms for the microarrays are phylochips, microbial diagnostic microarrays and identification arrays. Their principle is based on the dot-blot hybridization described above. Typically microarrays contain hundreds of oligonucleotide probes, usually based on the 16S rRNA gene, specific for different strains or species or genera of microorganisms that are detected in a single assay. Total DNA or RNA is isolated from the sample, fragmented, and amplified by PCR with the simultaneous incorporation of labeled nucleotides, or directly chemically labeled. The labeled fragments are hybridized to the probes immobilized on a surface, and following washing hybridized fragments are detected by a fluorescence scanner. There are many different forms of arrays to which the probes can be attached including macroarrays, and glass microarrays that are low to medium density, and very high density Affymetric microarrays (> 104 probes typically 25 mer per chip) (108). Three-dimensional form microarrays such as the Pamgene system and gel-pads allow the option for quantitative detection (109). Studies are underway to apply microarray technology to the human intestinal microbiota (16). A macroarray membrane-based method with 60 40-mer oligonucleotide probes specific for the dominant microbiota demonstrated the feasibility of arrays for detection (110). The high throughput potential of arrays will undoubtedly encourage further efforts in this area in the coming years.
Artificial Rumen

redpajama_set_name: RedPajamaCommonCrawl